Theorem cleqf 2279

Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2214. (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 2109	. 2 |- ( A = B <-> A. y ( y e. A <-> y e. B ) )
2		nfv 1491	. . 3 |- F/ y ( x e. A <-> x e. B )
3		cleqf.1	. . . . 5 |- F/_ x A
4	3	nfcri 2249	. . . 4 |- F/ x y e. A
5		cleqf.2	. . . . 5 |- F/_ x B
6	5	nfcri 2249	. . . 4 |- F/ x y e. B
7	4, 6	nfbi 1551	. . 3 |- F/ x ( y e. A <-> y e. B )
8		eleq1 2177	. . . 4 |- ( x = y -> ( x e. A <-> y e. A ) )
9		eleq1 2177	. . . 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 1710	. 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 1312   = wceq 1314   e. wcel 1463   F/_wnfc 2242

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-cleq 2108  df-clel 2111  df-nfc 2244

This theorem is referenced by:  abid2f  2280  n0rf  3341  eq0  3347  iunab  3825  iinab  3840  sniota  5073