Theorem rabeqif 2721

Description: Equality theorem for restricted class abstractions. Inference form of rabeqf 2720. (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 2720	. 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 1348  Ⅎwnfc 2299  {crab 2452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457

This theorem is referenced by:  rabeqi  2723