Theorem cleqf 2513

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

Hypotheses

Ref	Expression
cleqf.1	|- F/_xA
cleqf.2	|- F/_xB

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 2347	. 2 |- (A = B <-> A.y(y e. A <-> y e. B))
2		nfv 1619	. . 3 |- F/y(x e. A <-> x e. B)
3		cleqf.1	. . . . 5 |- F/_xA
4	3	nfcri 2483	. . . 4 |- F/x y e. A
5		cleqf.2	. . . . 5 |- F/_xB
6	5	nfcri 2483	. . . 4 |- F/x y e. B
7	4, 6	nfbi 1834	. . 3 |- F/x(y e. A <-> y e. B)
8		eleq1 2413	. . . 4 |- (x = y -> (x e. A <-> y e. A))
9		eleq1 2413	. . . 4 |- (x = y -> (x e. B <-> y e. B))
10	8, 9	bibi12d 312	. . 3 |- (x = y -> ((x e. A <-> x e. B) <-> (y e. A <-> y e. B)))
11	2, 7, 10	cbval 1984	. 2 |- (A.x(x e. A <-> x e. B) <-> A.y(y e. A <-> y e. B))
12	1, 11	bitr4i 243	1 |- (A = B <-> A.x(x e. A <-> x e. B))

Syntax hints:   <-> wb 176  A.wal 1540   = wceq 1642   e. wcel 1710   F/_wnfc 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478

This theorem is referenced by:  abid2f  2514  n0f  3558  iunab  4012  iinab  4027  sniota  4369