Theorem rabeqiOLD 3466

Description: Obsolete version of rabeqi 3440 as of 3-Jun-2024. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Avoid ax-10 2130 and ax-11 2147. (Revised by Gino Giotto, 20-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
rabeqiOLD.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
rabeqiOLD	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}

Proof of Theorem rabeqiOLD

Step	Hyp	Ref	Expression
1		rabeqiOLD.1	. 2 ⊢ 𝐴 = 𝐵
2	1	nfth 1796	. . . 4 ⊢ Ⅎ𝑥 𝐴 = 𝐵
3		eleq2 2817	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
4	3	anbi1d 629	. . . 4 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)))
5	2, 4	abbid 2798	. . 3 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)})
6		df-rab 3428	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
7		df-rab 3428	. . 3 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
8	5, 6, 7	3eqtr4g 2792	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑})
9	1, 8	ax-mp 5	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}

Syntax hints:   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {cab 2704  {crab 3427

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-12 2164  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3428

This theorem is referenced by: (None)