Theorem abbi 2319

Description: Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.)

Assertion

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

Proof of Theorem abbi

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2199	. 2 |- ( { x | ph } = { x | ps } <-> A. y ( y e. { x | ph } <-> y e. { x | ps } ) )
2		nfsab1 2195	. . . 4 |- F/ x y e. { x | ph }
3		nfsab1 2195	. . . 4 |- F/ x y e. { x | ps }
4	2, 3	nfbi 1612	. . 3 |- F/ x ( y e. { x | ph } <-> y e. { x | ps } )
5		nfv 1551	. . 3 |- F/ y ( ph <-> ps )
6		df-clab 2192	. . . . 5 |- ( y e. { x | ph } <-> [ y / x ] ph )
7		sbequ12r 1795	. . . . 5 |- ( y = x -> ( [ y / x ] ph <-> ph ) )
8	6, 7	bitrid 192	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> ph ) )
9		df-clab 2192	. . . . 5 |- ( y e. { x | ps } <-> [ y / x ] ps )
10		sbequ12r 1795	. . . . 5 |- ( y = x -> ( [ y / x ] ps <-> ps ) )
11	9, 10	bitrid 192	. . . 4 |- ( y = x -> ( y e. { x | ps } <-> ps ) )
12	8, 11	bibi12d 235	. . 3 |- ( y = x -> ( ( y e. { x | ph } <-> y e. { x | ps } ) <-> ( ph <-> ps ) ) )
13	4, 5, 12	cbval 1777	. 2 |- ( A. y ( y e. { x | ph } <-> y e. { x | ps } ) <-> A. x ( ph <-> ps ) )
14	1, 13	bitr2i 185	1 |- ( A. x ( ph <-> ps ) <-> { x | ph } = { x | ps } )

Syntax hints:   <-> wb 105   A.wal 1371   = wceq 1373   [wsb 1785   e. wcel 2176   {cab 2191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198

This theorem is referenced by:  abbii  2321  abbid  2322  rabbi  2684  sbcbi2  3049  dfiota2  5234  iotabi  5242  uniabio  5243