Burhān al-Ṣiddīqīn (Arabic: برهان الصّدّیقین) is a well-known theological argument for the existence of God which was mentioned for the first time by al-Farabi. Later, Avicenna introduced a new approach towards this argument and called it Burhan al-Siddiqin.
Unlike the teleological or cosmological argument, in this argument the existence of God is not proven through the mediation of anything (middle-terms such as: movement, creation, possibility, etc.) other than God himself. All that is used are mediating entities that are identical with God, such as entities like reality, existence, being, necessity, and so forth.
Indeed, we see a variety of versions of the Burhan al-Siddiqin in Islamic philosophy. Avicenna, al-Suhrawardi, and Mulla Sadra are among classical philosophers who, after al-Farabi, have expressed some new versions of the argument. Likewise, Anselm's ontological argument is very similar to this kind of argument but he founds his argument upon a conceptual definition.
Three major explanations of this argument are as follows.
Avicenna divides beings into necessary and contingent beings. Then, he states that a contingent being could not come into existence by itself but it must depend upon another being in its existence.
Now, what is another being itself? If it would be a necessary being, so it is our favorable conclusion; but if it would be a contingent being, we will be faced with the same question again and it ends up going into infinite regress or a vicious circle, unless we accept a necessary being. Through these means, he proves the existence of God as the necessary being.
Mulla Sadra's Explanation
He believes in the reality of the existence and the impossibility of its inexistence. Likewise, he states this existence is free of any condition and limitation in its essence. Reality exists just because it exists, neither because of some assumptions nor something else. So, reality in its essence is equal to being independent of others and it exists unconditionally. And it equals to eternal necessary being. Therefore, reality of existence in its essence –and in itself without any conditions is equal to the eternal essence of God. So, the priority of existence leads us directly to the eternal essence of God.
Muhammad Husayn Tabataba'i's Explanation
Tabataba'i in his commentary on the book of al-Asfar by Mulla Sadra presents his explanation of the argument. Indeed, his argument is independent of any medium (middle terms) that is independent of any philosophical principles and presuppositions, even the priority of existence. He says: before any discussion about a two-cornered debate of the priority of existence/the priority of essence, we admit reality in itself and it is the point of separation from the skepticism. We deny skepticism and find that every reasonable individual inevitably accepts it.
Then, he states that this reality is one that cannot adopt nonexistence and nullity in its essence. Even the supposition of the nonexistence and nullity of reality itself necessitates its existence. In other words, if we suppose that all realities are invalid and inexistence in a particular time or always, this means that in reality, all realities are nonexistent (and this in turn will prove the existence of a reality again). Likewise, if the sophist assumes that all things are illusionary, or doubts in their reality, indeed in his view, those things are really illusionary and their realities are really dubitable (this means that reality is proven from the very point it was rejected), while reality in essence, and cannot adopt any inexistence and nullity, resulting in its necessity by itself. And this necessary essence is equal to eternal essence. Therefore, there is a reality that is eternal and necessary by itself that makes other things that are real, dependent on it in their reality and existence. It is from this that the thinker finds that the existence of an eternal and necessary being through itself is obvious for everyone and the proofs for the existence of the necessary being are actually nothing but reminders.
Quranic and Hadith Grounds
Burhan al-Siddiqin, which is presented in theological context with philosophical methods, is thoroughly adopted with Quranic thoughts, Islamic hadiths, and mysticism. Mulla Sadra says he was inspired by mystical intuition in forming this proof. And we see many versus of the Qu'an and some hadith in all versions of the argument with excellent evidential forces. Some of them are mentioned here:
- Soon We shall show them Our signs in the horizons and in their own souls until it becomes clear to them that He is the real. Is it not sufficient that your Lord is witness to all things? 
- I recognized you by you!
- Oh, who signifies to its essence by its essence!
- How could someone justifies your existence based on something which depends on you in its existence!?
- Do other things have some kind of appearance or clarification which you lack, till they clarify you?
- Qur'an 41:53: سَنُريهِم آياتِنا فِي الآفاقِ وَفي أَنفُسِهِم حَتّىٰ يَتَبَيَّنَ لَهُم أَنَّهُ الحَقُّ أَوَلَم يَكفِ بِرَبِّكَ أَنَّهُ عَلىٰ كُلِّ شَيءٍ شَهيدٌ
- The material for this article is mainly taken from برهان صدیقین in Farsi WikiShia.