Theorem cleqf 2242

Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2178. (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 2075	. 2 |- ( A = B <-> A. y ( y e. A <-> y e. B ) )
2		nfv 1461	. . 3 |- F/ y ( x e. A <-> x e. B )
3		cleqf.1	. . . . 5 |- F/_ x A
4	3	nfcri 2213	. . . 4 |- F/ x y e. A
5		cleqf.2	. . . . 5 |- F/_ x B
6	5	nfcri 2213	. . . 4 |- F/ x y e. B
7	4, 6	nfbi 1521	. . 3 |- F/ x ( y e. A <-> y e. B )
8		eleq1 2141	. . . 4 |- ( x = y -> ( x e. A <-> y e. A ) )
9		eleq1 2141	. . . 4 |- ( x = y -> ( x e. B <-> y e. B ) )
10	8, 9	bibi12d 233	. . 3 |- ( x = y -> ( ( x e. A <-> x e. B ) <-> ( y e. A <-> y e. B ) ) )
11	2, 7, 10	cbval 1677	. 2 |- ( A. x ( x e. A <-> x e. B ) <-> A. y ( y e. A <-> y e. B ) )
12	1, 11	bitr4i 185	1 |- ( A = B <-> A. x ( x e. A <-> x e. B ) )

Syntax hints:   <-> wb 103   A.wal 1282   = wceq 1284   e. wcel 1433   F/_wnfc 2206

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-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  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-cleq 2074  df-clel 2077  df-nfc 2208

This theorem is referenced by:  abid2f  2243  n0rf  3260  eq0  3266  iunab  3724  iinab  3739  sniota  4914