Theorem cleqf 2373

Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2305. (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 2199	. 2 |- ( A = B <-> A. y ( y e. A <-> y e. B ) )
2		nfv 1551	. . 3 |- F/ y ( x e. A <-> x e. B )
3		cleqf.1	. . . . 5 |- F/_ x A
4	3	nfcri 2342	. . . 4 |- F/ x y e. A
5		cleqf.2	. . . . 5 |- F/_ x B
6	5	nfcri 2342	. . . 4 |- F/ x y e. B
7	4, 6	nfbi 1612	. . 3 |- F/ x ( y e. A <-> y e. B )
8		eleq1 2268	. . . 4 |- ( x = y -> ( x e. A <-> y e. A ) )
9		eleq1 2268	. . . 4 |- ( x = y -> ( x e. B <-> y e. B ) )
10	8, 9	bibi12d 235	. . 3 |- ( x = y -> ( ( x e. A <-> x e. B ) <-> ( y e. A <-> y e. B ) ) )
11	2, 7, 10	cbval 1777	. 2 |- ( A. x ( x e. A <-> x e. B ) <-> A. y ( y e. A <-> y e. B ) )
12	1, 11	bitr4i 187	1 |- ( A = B <-> A. x ( x e. A <-> x e. B ) )

Syntax hints:   <-> wb 105   A.wal 1371   = wceq 1373   e. wcel 2176   F/_wnfc 2335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-cleq 2198  df-clel 2201  df-nfc 2337

This theorem is referenced by:  abid2f  2374  n0rf  3473  eq0  3479  iunab  3974  iinab  3989  sniota  5262