Theorem rabeqi 2728

Description: Equality theorem for restricted class abstractions. Inference form of rabeq 2727. (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 2317	. 2 |- F/_ x A
2		nfcv 2317	. 2 |- F/_ x B
3		rabeqi.1	. 2 |- A = B
4	1, 2, 3	rabeqif 2726	1 |- { x e. A | ph } = { x e. B | ph }

Syntax hints:   = wceq 1353   {crab 2457

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rab 2462

This theorem is referenced by:  phimullem  12192  odzcllem  12209  odzdvds  12212