Theorem rabeqif 2672

Description: Equality theorem for restricted class abstractions. Inference form of rabeqf 2671. (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 2671	. 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 1331   F/_wnfc 2266   {crab 2418

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rab 2423

This theorem is referenced by:  rabeqi  2674