Theorem rabeqif 3191

Description: Equality theorem for restricted class abstractions. Inference form of rabeqf 3190. (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 3190	. 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 1483   F/_wnfc 2751   {crab 2916

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921

This theorem is referenced by:  rabeqi  3193  smfliminf  41037