Theorem rabeqif 2721

Description: Equality theorem for restricted class abstractions. Inference form of rabeqf 2720. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
rabeqf.1	|- F/_ x A
rabeqf.2	|- F/_ x B
rabeqif.3	|- A = B

Assertion

Ref	Expression
rabeqif	|- { x e. A | ph } = { x e. B | ph }

Proof of Theorem rabeqif

Step	Hyp	Ref	Expression
1		rabeqif.3	. 2 |- A = B
2		rabeqf.1	. . 3 |- F/_ x A
3		rabeqf.2	. . 3 |- F/_ x B
4	2, 3	rabeqf 2720	. 2 |- ( A = B -> { x e. A | ph } = { x e. B | ph } )
5	1, 4	ax-mp 5	1 |- { x e. A | ph } = { x e. B | ph }

Syntax hints:   = wceq 1348   F/_wnfc 2299   {crab 2452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457

This theorem is referenced by:  rabeqi  2723