Theorem rabeqf 3463

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	. . . 4 ⊢ Ⅎ𝑥𝐴
2		rabeqf.2	. . . 4 ⊢ Ⅎ𝑥𝐵
3	1, 2	nfeq 2913	. . 3 ⊢ Ⅎ𝑥 𝐴 = 𝐵
4		eleq2 2818	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
5	4	anbi1d 630	. . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)))
6	3, 5	abbid 2799	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)})
7		df-rab 3430	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
8		df-rab 3430	. 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
9	6, 7, 8	3eqtr4g 2793	1 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {cab 2705  Ⅎwnfc 2879  {crab 3429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-rab 3430

This theorem is referenced by:  fpwrelmapffs  32529  rabeq12f  37630  issmfdf  46125  smfpimltmpt  46134  smfpimltxrmptf  46146  smfpimgtmpt  46169  smfpimgtxrmptf  46172  smfsupmpt  46203  smfinfmpt  46207