Theorem abbi 2192

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 2075	. 2 |- ( { x | ph } = { x | ps } <-> A. y ( y e. { x | ph } <-> y e. { x | ps } ) )
2		nfsab1 2071	. . . 4 |- F/ x y e. { x | ph }
3		nfsab1 2071	. . . 4 |- F/ x y e. { x | ps }
4	2, 3	nfbi 1521	. . 3 |- F/ x ( y e. { x | ph } <-> y e. { x | ps } )
5		nfv 1461	. . 3 |- F/ y ( ph <-> ps )
6		df-clab 2068	. . . . 5 |- ( y e. { x | ph } <-> [ y / x ] ph )
7		sbequ12r 1695	. . . . 5 |- ( y = x -> ( [ y / x ] ph <-> ph ) )
8	6, 7	syl5bb 190	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> ph ) )
9		df-clab 2068	. . . . 5 |- ( y e. { x | ps } <-> [ y / x ] ps )
10		sbequ12r 1695	. . . . 5 |- ( y = x -> ( [ y / x ] ps <-> ps ) )
11	9, 10	syl5bb 190	. . . 4 |- ( y = x -> ( y e. { x | ps } <-> ps ) )
12	8, 11	bibi12d 233	. . 3 |- ( y = x -> ( ( y e. { x | ph } <-> y e. { x | ps } ) <-> ( ph <-> ps ) ) )
13	4, 5, 12	cbval 1677	. 2 |- ( A. y ( y e. { x | ph } <-> y e. { x | ps } ) <-> A. x ( ph <-> ps ) )
14	1, 13	bitr2i 183	1 |- ( A. x ( ph <-> ps ) <-> { x | ph } = { x | ps } )

Syntax hints:   <-> wb 103   A.wal 1282   = wceq 1284   e. wcel 1433   [wsb 1685   {cab 2067

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074

This theorem is referenced by:  abbii  2194  abbid  2195  rabbi  2531  sbcbi2  2864  dfiota2  4888  iotabi  4896  uniabio  4897