Theorem rabeqi 2756

Description: Equality theorem for restricted class abstractions. Inference form of rabeq 2755. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
rabeqi.1	|- A = B

Assertion

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

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem rabeqi

Step	Hyp	Ref	Expression
1		nfcv 2339	. 2 |- F/_ x A
2		nfcv 2339	. 2 |- F/_ x B
3		rabeqi.1	. 2 |- A = B
4	1, 2, 3	rabeqif 2754	1 |- { x e. A | ph } = { x e. B | ph }

Syntax hints:   = wceq 1364   {crab 2479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484

This theorem is referenced by:  bitsfzolem  12118  lcmval  12231  lcmcllem  12235  lcmledvds  12238  phimullem  12393  odzcllem  12411  odzdvds  12414  4sqlem13m  12572  4sqlem14  12573  4sqlem17  12576  4sqlem18  12577