Theorem bj-abbi 32775

Description: Remove dependency on ax-13 2246 from abbi 2737. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem bj-abbi

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2616	. 2 |- ( { x | ph } = { x | ps } <-> A. y ( y e. { x | ph } <-> y e. { x | ps } ) )
2		bj-nfsab1 32772	. . . 4 |- F/ x y e. { x | ph }
3		bj-nfsab1 32772	. . . 4 |- F/ x y e. { x | ps }
4	2, 3	nfbi 1833	. . 3 |- F/ x ( y e. { x | ph } <-> y e. { x | ps } )
5		nfv 1843	. . 3 |- F/ y ( ph <-> ps )
6		df-clab 2609	. . . . 5 |- ( y e. { x | ph } <-> [ y / x ] ph )
7		sbequ12r 2112	. . . . 5 |- ( y = x -> ( [ y / x ] ph <-> ph ) )
8	6, 7	syl5bb 272	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> ph ) )
9		df-clab 2609	. . . . 5 |- ( y e. { x | ps } <-> [ y / x ] ps )
10		sbequ12r 2112	. . . . 5 |- ( y = x -> ( [ y / x ] ps <-> ps ) )
11	9, 10	syl5bb 272	. . . 4 |- ( y = x -> ( y e. { x | ps } <-> ps ) )
12	8, 11	bibi12d 335	. . 3 |- ( y = x -> ( ( y e. { x | ph } <-> y e. { x | ps } ) <-> ( ph <-> ps ) ) )
13	4, 5, 12	cbvalv1 2175	. 2 |- ( A. y ( y e. { x | ph } <-> y e. { x | ps } ) <-> A. x ( ph <-> ps ) )
14	1, 13	bitr2i 265	1 |- ( A. x ( ph <-> ps ) <-> { x | ph } = { x | ps } )

Syntax hints:   <-> wb 196   A.wal 1481   = wceq 1483   [wsb 1880   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

This theorem is referenced by:  bj-abbii  32777  bj-abbid  32778