Full Idea
Comprehension Scheme: for each formula φ without y free, the universal closure of this is an axiom: ∃y ∀x (x ∈ y ↔ x ∈ z ∧ φ). That is, there must be a set y if it can be defined by the formula φ.
Gist of Idea
Comprehension: ∃y ∀x (x ∈ y ↔ x ∈ z ∧ φ)
Source
Kenneth Kunen (Set Theory [1980], §1.5)
Book Reference
Kunen,Kenneth: 'Set Theory: Introduction to Independence Proofs' [North-Holland 1980], p.11
A Reaction
Unrestricted comprehension leads to Russell's paradox, so restricting it in some way (e.g. by the Axiom of Specification) is essential.