Theorem bj-abeq2 32773

Description: Remove dependency on ax-13 2246 from abeq2 2732. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-abeq2	|- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem bj-abeq2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-5 1839	. . 3 |- ( y e. A -> A. x y e. A )
2		bj-hbab1 32771	. . 3 |- ( y e. { x | ph } -> A. x y e. { x | ph } )
3	1, 2	cleqh 2724	. 2 |- ( A = { x | ph } <-> A. x ( x e. A <-> x e. { x | ph } ) )
4		abid 2610	. . . 4 |- ( x e. { x | ph } <-> ph )
5	4	bibi2i 327	. . 3 |- ( ( x e. A <-> x e. { x | ph } ) <-> ( x e. A <-> ph ) )
6	5	albii 1747	. 2 |- ( A. x ( x e. A <-> x e. { x | ph } ) <-> A. x ( x e. A <-> ph ) )
7	3, 6	bitri 264	1 |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) )

Syntax hints:   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618

This theorem is referenced by:  bj-abeq1  32774  bj-abbi2i  32776  bj-abbi2dv  32780  bj-clabel  32783  bj-ru1  32933