Theorem cleqf 2331

Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2264. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)

Hypotheses

Ref	Expression
cleqf.1	|- F/_ x A
cleqf.2	|- F/_ x B

Assertion

Ref	Expression
cleqf	|- ( A = B <-> A. x ( x e. A <-> x e. B ) )

Proof of Theorem cleqf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2158	. 2 |- ( A = B <-> A. y ( y e. A <-> y e. B ) )
2		nfv 1515	. . 3 |- F/ y ( x e. A <-> x e. B )
3		cleqf.1	. . . . 5 |- F/_ x A
4	3	nfcri 2300	. . . 4 |- F/ x y e. A
5		cleqf.2	. . . . 5 |- F/_ x B
6	5	nfcri 2300	. . . 4 |- F/ x y e. B
7	4, 6	nfbi 1576	. . 3 |- F/ x ( y e. A <-> y e. B )
8		eleq1 2227	. . . 4 |- ( x = y -> ( x e. A <-> y e. A ) )
9		eleq1 2227	. . . 4 |- ( x = y -> ( x e. B <-> y e. B ) )
10	8, 9	bibi12d 234	. . 3 |- ( x = y -> ( ( x e. A <-> x e. B ) <-> ( y e. A <-> y e. B ) ) )
11	2, 7, 10	cbval 1741	. 2 |- ( A. x ( x e. A <-> x e. B ) <-> A. y ( y e. A <-> y e. B ) )
12	1, 11	bitr4i 186	1 |- ( A = B <-> A. x ( x e. A <-> x e. B ) )

Syntax hints:   <-> wb 104   A.wal 1340   = wceq 1342   e. wcel 2135   F/_wnfc 2293

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-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-cleq 2157  df-clel 2160  df-nfc 2295

This theorem is referenced by:  abid2f  2332  n0rf  3416  eq0  3422  iunab  3906  iinab  3921  sniota  5174