Theorem abbi 2284

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 2164	. 2 |- ( { x | ph } = { x | ps } <-> A. y ( y e. { x | ph } <-> y e. { x | ps } ) )
2		nfsab1 2160	. . . 4 |- F/ x y e. { x | ph }
3		nfsab1 2160	. . . 4 |- F/ x y e. { x | ps }
4	2, 3	nfbi 1582	. . 3 |- F/ x ( y e. { x | ph } <-> y e. { x | ps } )
5		nfv 1521	. . 3 |- F/ y ( ph <-> ps )
6		df-clab 2157	. . . . 5 |- ( y e. { x | ph } <-> [ y / x ] ph )
7		sbequ12r 1765	. . . . 5 |- ( y = x -> ( [ y / x ] ph <-> ph ) )
8	6, 7	syl5bb 191	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> ph ) )
9		df-clab 2157	. . . . 5 |- ( y e. { x | ps } <-> [ y / x ] ps )
10		sbequ12r 1765	. . . . 5 |- ( y = x -> ( [ y / x ] ps <-> ps ) )
11	9, 10	syl5bb 191	. . . 4 |- ( y = x -> ( y e. { x | ps } <-> ps ) )
12	8, 11	bibi12d 234	. . 3 |- ( y = x -> ( ( y e. { x | ph } <-> y e. { x | ps } ) <-> ( ph <-> ps ) ) )
13	4, 5, 12	cbval 1747	. 2 |- ( A. y ( y e. { x | ph } <-> y e. { x | ps } ) <-> A. x ( ph <-> ps ) )
14	1, 13	bitr2i 184	1 |- ( A. x ( ph <-> ps ) <-> { x | ph } = { x | ps } )

Syntax hints:   <-> wb 104   A.wal 1346   = wceq 1348   [wsb 1755   e. wcel 2141   {cab 2156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163

This theorem is referenced by:  abbii  2286  abbid  2287  rabbi  2647  sbcbi2  3005  dfiota2  5161  iotabi  5169  uniabio  5170