Theorem abbi 2251

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 2131	. 2 |- ( { x | ph } = { x | ps } <-> A. y ( y e. { x | ph } <-> y e. { x | ps } ) )
2		nfsab1 2127	. . . 4 |- F/ x y e. { x | ph }
3		nfsab1 2127	. . . 4 |- F/ x y e. { x | ps }
4	2, 3	nfbi 1568	. . 3 |- F/ x ( y e. { x | ph } <-> y e. { x | ps } )
5		nfv 1508	. . 3 |- F/ y ( ph <-> ps )
6		df-clab 2124	. . . . 5 |- ( y e. { x | ph } <-> [ y / x ] ph )
7		sbequ12r 1745	. . . . 5 |- ( y = x -> ( [ y / x ] ph <-> ph ) )
8	6, 7	syl5bb 191	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> ph ) )
9		df-clab 2124	. . . . 5 |- ( y e. { x | ps } <-> [ y / x ] ps )
10		sbequ12r 1745	. . . . 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 1727	. 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 1329   = wceq 1331   e. wcel 1480   [wsb 1735   {cab 2123

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130

This theorem is referenced by:  abbii  2253  abbid  2254  rabbi  2606  sbcbi2  2954  dfiota2  5084  iotabi  5092  uniabio  5093