Theorem rabeqif 2812

Description: Equality theorem for restricted class abstractions. Inference form of rabeqf 2811. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
rabeqf.1	⊢ Ⅎ𝑥𝐴
rabeqf.2	⊢ Ⅎ𝑥𝐵
rabeqif.3	⊢ 𝐴 = 𝐵

Assertion

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

Proof of Theorem rabeqif

Step	Hyp	Ref	Expression
1		rabeqif.3	. 2 ⊢ 𝐴 = 𝐵
2		rabeqf.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		rabeqf.2	. . 3 ⊢ Ⅎ𝑥𝐵
4	2, 3	rabeqf 2811	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑})
5	1, 4	ax-mp 5	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}

Syntax hints:   = wceq 1402  Ⅎwnfc 2379  {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:  rabeqi  2814