Theorem rabeqif 3404

Description: Equality theorem for restricted class abstractions. Inference form of rabeqf 3403. (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 3403	. 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 1656  Ⅎwnfc 2956  {crab 3121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rab 3126

This theorem is referenced by:  rabeqi  3406  rabrabi  3413  extwwlkfab  27736  extwwlkfabOLD  27737  smfliminf  41825