Theorem cleqf 2790

Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2724. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.)

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		cleqf.1	. . 3 |- F/_ x A
2	1	nfcrii 2757	. 2 |- ( y e. A -> A. x y e. A )
3		cleqf.2	. . 3 |- F/_ x B
4	3	nfcrii 2757	. 2 |- ( y e. B -> A. x y e. B )
5	2, 4	cleqh 2724	1 |- ( A = B <-> A. x ( x e. A <-> x e. B ) )

Syntax hints:   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990   F/_wnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-cleq 2615  df-clel 2618  df-nfc 2753

This theorem is referenced by:  abid2f  2791  eqvf  3204  eqrd  3622  eq0f  3925  n0fOLD  3928  iunab  4566  iinab  4581  mbfposr  23419  mbfinf  23432  itg1climres  23481  bnj1366  30900  bj-rabtrALT  32927  compab  38645  dfcleqf  39255