Theorem rabeqi 2742

Description: Equality theorem for restricted class abstractions. Inference form of rabeq 2741. (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 2329	. 2 |- F/_ x A
2		nfcv 2329	. 2 |- F/_ x B
3		rabeqi.1	. 2 |- A = B
4	1, 2, 3	rabeqif 2740	1 |- { x e. A | ph } = { x e. B | ph }

Syntax hints:   = wceq 1363   {crab 2469

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rab 2474

This theorem is referenced by:  phimullem  12238  odzcllem  12255  odzdvds  12258