Theorem rabeqi 2769

Description: Equality theorem for restricted class abstractions. Inference form of rabeq 2768. (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 2350	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2350	. 2 ⊢ Ⅎ𝑥𝐵
3		rabeqi.1	. 2 ⊢ 𝐴 = 𝐵
4	1, 2, 3	rabeqif 2767	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}

Syntax hints:   = wceq 1373  {crab 2490

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rab 2495

This theorem is referenced by:  bitsfzolem  12380  lcmval  12500  lcmcllem  12504  lcmledvds  12507  phimullem  12662  odzcllem  12680  odzdvds  12683  4sqlem13m  12841  4sqlem14  12842  4sqlem17  12845  4sqlem18  12846  pw0ss  15794