Theorem rabeqf 3472

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 2919	. . 3 ⊢ Ⅎ𝑥 𝐴 = 𝐵
4		eleq2 2830	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
5	4	anbi1d 631	. . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)))
6	3, 5	abbid 2810	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)})
7		df-rab 3437	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
8		df-rab 3437	. 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
9	6, 7, 8	3eqtr4g 2802	1 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  Ⅎwnfc 2890  {crab 3436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437

This theorem is referenced by:  fpwrelmapffs  32745  rabeq12f  38164  issmfdf  46752  smfpimltmpt  46761  smfpimltxrmptf  46773  smfpimgtmpt  46796  smfpimgtxrmptf  46799  smfsupmpt  46830