Theorem rabeqf 3438

Description: Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.)

Hypotheses

Ref	Expression
rabeqf.1	⊢ Ⅎ𝑥𝐴
rabeqf.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
rabeqf	⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑})

Proof of Theorem rabeqf

Step	Hyp	Ref	Expression
1		rabeqf.1	. . 3 ⊢ Ⅎ𝑥𝐴
2		rabeqf.2	. . 3 ⊢ Ⅎ𝑥𝐵
3	1, 2	nfeq 2927	. 2 ⊢ Ⅎ𝑥 𝐴 = 𝐵
4		eleq2 2841	. . 3 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
5	4	anbi1d 639	. 2 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)))
6	3, 5	rabbida4 3429	1 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1550   ∈ wcel 2132  Ⅎwnfc 2899  {crab 3404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-tru 1553  df-ex 1790  df-nf 1794  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-rab 3405

This theorem is referenced by:  fpwrelmapffs  32875  rabeq12f  38594  issmfdf  47249  smfpimltmpt  47258  smfpimltxrmptf  47270  smfpimgtmpt  47293  smfpimgtxrmptf  47296  smfsupmpt  47327