Theorem abbi 2310

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 2190	. 2 |- ( { x | ph } = { x | ps } <-> A. y ( y e. { x | ph } <-> y e. { x | ps } ) )
2		nfsab1 2186	. . . 4 |- F/ x y e. { x | ph }
3		nfsab1 2186	. . . 4 |- F/ x y e. { x | ps }
4	2, 3	nfbi 1603	. . 3 |- F/ x ( y e. { x | ph } <-> y e. { x | ps } )
5		nfv 1542	. . 3 |- F/ y ( ph <-> ps )
6		df-clab 2183	. . . . 5 |- ( y e. { x | ph } <-> [ y / x ] ph )
7		sbequ12r 1786	. . . . 5 |- ( y = x -> ( [ y / x ] ph <-> ph ) )
8	6, 7	bitrid 192	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> ph ) )
9		df-clab 2183	. . . . 5 |- ( y e. { x | ps } <-> [ y / x ] ps )
10		sbequ12r 1786	. . . . 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 1768	. 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 1362   = wceq 1364   [wsb 1776   e. wcel 2167   {cab 2182

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189

This theorem is referenced by:  abbii  2312  abbid  2313  rabbi  2675  sbcbi2  3040  dfiota2  5220  iotabi  5228  uniabio  5229