Theorem rabeqi 2795

Description: Equality theorem for restricted class abstractions. Inference form of rabeq 2794. (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 2374	. 2 |- F/_ x A
2		nfcv 2374	. 2 |- F/_ x B
3		rabeqi.1	. 2 |- A = B
4	1, 2, 3	rabeqif 2793	1 |- { x e. A | ph } = { x e. B | ph }

Syntax hints:   = wceq 1397   {crab 2514

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519

This theorem is referenced by:  bitsfzolem  12514  lcmval  12634  lcmcllem  12638  lcmledvds  12641  phimullem  12796  odzcllem  12814  odzdvds  12817  4sqlem13m  12975  4sqlem14  12976  4sqlem17  12979  4sqlem18  12980  pw0ss  15933