Theorem rabeqi 2814

Description: Equality theorem for restricted class abstractions. Inference form of rabeq 2813. (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 2392	. 2 |- F/_ x A
2		nfcv 2392	. 2 |- F/_ x B
3		rabeqi.1	. 2 |- A = B
4	1, 2, 3	rabeqif 2812	1 |- { x e. A | ph } = { x e. B | ph }

Syntax hints:   = wceq 1402   {crab 2532

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537

This theorem is referenced by:  hashfibc  11261  bitsfzolem  12699  lcmval  12819  lcmcllem  12823  lcmledvds  12826  phimullem  12981  odzcllem  12999  odzdvds  13002  4sqlem13m  13160  4sqlem14  13161  4sqlem17  13164  4sqlem18  13165  pw0ss  16238  konigsbergiedgwen  16639