Theorem eqabdv

Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994) Avoid ax-11 . (Revised by Wolf Lammen, 6-May-2023)

	Ref	Expression
Hypothesis	eqabdv.1	$$⊢\phi \to \left(x\in A\leftrightarrow \psi \right)$$
Assertion	eqabdv	$$⊢\phi \to A=\left\{x|\psi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		eqabdv.1	$$⊢\phi \to \left(x\in A\leftrightarrow \psi \right)$$
2	1	sbbidv	$$⊢\phi \to \left(\left[y/x\right]x\in A\leftrightarrow \left[y/x\right]\psi \right)$$
3		clelsb1	$$⊢\left[y/x\right]x\in A\leftrightarrow y\in A$$
4	3	bicomi	$$⊢y\in A\leftrightarrow \left[y/x\right]x\in A$$
5		df-clab	$$⊢y\in \left\{x|\psi \right\}\leftrightarrow \left[y/x\right]\psi$$
6	2 4 5	3bitr4g	$$⊢\phi \to \left(y\in A\leftrightarrow y\in \left\{x|\psi \right\}\right)$$
7	6	eqrdv	$$⊢\phi \to A=\left\{x|\psi \right\}$$