Theorem rabeqif 2728

Description: Equality theorem for restricted class abstractions. Inference form of rabeqf 2727. (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 2727	. 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 1353  Ⅎwnfc 2306  {crab 2459

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464

This theorem is referenced by:  rabeqi  2730