Theorem eqri 29315

Description: Infer equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 7-Oct-2017.)

Hypotheses

Ref	Expression
eqri.1	|- F/_ x A
eqri.2	|- F/_ x B
eqri.3	|- ( x e. A <-> x e. B )

Assertion

Ref	Expression
eqri	|- A = B

Proof of Theorem eqri

Step	Hyp	Ref	Expression
1		nftru 1730	. . 3 |- F/ x T.
2		eqri.1	. . 3 |- F/_ x A
3		eqri.2	. . 3 |- F/_ x B
4		eqri.3	. . . 4 |- ( x e. A <-> x e. B )
5	4	a1i 11	. . 3 |- ( T. -> ( x e. A <-> x e. B ) )
6	1, 2, 3, 5	eqrd 3622	. 2 |- ( T. -> A = B )
7	6	trud 1493	1 |- A = B

Syntax hints:   <-> wb 196   = wceq 1483   T. wtru 1484   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:  difrab2  29339  esum2dlem  30154  eulerpartlemn  30443