Theorem rabeqi 2793

Description: Equality theorem for restricted class abstractions. Inference form of rabeq 2792. (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 2372	. 2 |- F/_ x A
2		nfcv 2372	. 2 |- F/_ x B
3		rabeqi.1	. 2 |- A = B
4	1, 2, 3	rabeqif 2791	1 |- { x e. A | ph } = { x e. B | ph }

Syntax hints:   = wceq 1395   {crab 2512

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517

This theorem is referenced by:  bitsfzolem  12505  lcmval  12625  lcmcllem  12629  lcmledvds  12632  phimullem  12787  odzcllem  12805  odzdvds  12808  4sqlem13m  12966  4sqlem14  12967  4sqlem17  12970  4sqlem18  12971  pw0ss  15924