Theorem eqabbOLD

Description: Obsolete version of eqabb as of 12-Feb-2025. (Contributed by NM, 26-May-1993) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	eqabbOLD	$$⊢A=\left\{x|\phi \right\}\leftrightarrow \forall x\left(x\in A\leftrightarrow \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-5	$$⊢y\in A\to \forall xy\in A$$
2		hbab1	$$⊢y\in \left\{x|\phi \right\}\to \forall xy\in \left\{x|\phi \right\}$$
3	1 2	cleqh	$$⊢A=\left\{x|\phi \right\}\leftrightarrow \forall x\left(x\in A\leftrightarrow x\in \left\{x|\phi \right\}\right)$$
4		abid	$$⊢x\in \left\{x|\phi \right\}\leftrightarrow \phi$$
5	4	bibi2i	$$⊢\left(x\in A\leftrightarrow x\in \left\{x|\phi \right\}\right)\leftrightarrow \left(x\in A\leftrightarrow \phi \right)$$
6	5	albii	$$⊢\forall x\left(x\in A\leftrightarrow x\in \left\{x|\phi \right\}\right)\leftrightarrow \forall x\left(x\in A\leftrightarrow \phi \right)$$
7	3 6	bitri	$$⊢A=\left\{x|\phi \right\}\leftrightarrow \forall x\left(x\in A\leftrightarrow \phi \right)$$