Theorem rabeqi 2753

Description: Equality theorem for restricted class abstractions. Inference form of rabeq 2752. (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 2336	. 2 |- F/_ x A
2		nfcv 2336	. 2 |- F/_ x B
3		rabeqi.1	. 2 |- A = B
4	1, 2, 3	rabeqif 2751	1 |- { x e. A | ph } = { x e. B | ph }

Syntax hints:   = wceq 1364   {crab 2476

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481

This theorem is referenced by:  lcmval  12201  lcmcllem  12205  lcmledvds  12208  phimullem  12363  odzcllem  12380  odzdvds  12383  4sqlem13m  12541  4sqlem14  12542  4sqlem17  12545  4sqlem18  12546