Theorem cleqf 2357

Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2289. (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 2183	. 2 |- ( A = B <-> A. y ( y e. A <-> y e. B ) )
2		nfv 1539	. . 3 |- F/ y ( x e. A <-> x e. B )
3		cleqf.1	. . . . 5 |- F/_ x A
4	3	nfcri 2326	. . . 4 |- F/ x y e. A
5		cleqf.2	. . . . 5 |- F/_ x B
6	5	nfcri 2326	. . . 4 |- F/ x y e. B
7	4, 6	nfbi 1600	. . 3 |- F/ x ( y e. A <-> y e. B )
8		eleq1 2252	. . . 4 |- ( x = y -> ( x e. A <-> y e. A ) )
9		eleq1 2252	. . . 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 1765	. 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 1362   = wceq 1364   e. wcel 2160   F/_wnfc 2319

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-cleq 2182  df-clel 2185  df-nfc 2321

This theorem is referenced by:  abid2f  2358  n0rf  3450  eq0  3456  iunab  3948  iinab  3963  sniota  5226