Theorem rabeqi 2745

Description: Equality theorem for restricted class abstractions. Inference form of rabeq 2744. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
rabeqi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
rabeqi	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabeqi

Step	Hyp	Ref	Expression
1		nfcv 2332	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2332	. 2 ⊢ Ⅎ𝑥𝐵
3		rabeqi.1	. 2 ⊢ 𝐴 = 𝐵
4	1, 2, 3	rabeqif 2743	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}

Syntax hints:   = wceq 1364  {crab 2472

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 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477

This theorem is referenced by:  lcmval  12095  lcmcllem  12099  lcmledvds  12102  phimullem  12257  odzcllem  12274  odzdvds  12277  4sqlem13m  12435  4sqlem14  12436  4sqlem17  12439  4sqlem18  12440