Theorem rabeqbii 36173

Description: Equality theorem for restricted class abstractions. Inference version. (Contributed by GG, 1-Sep-2025.)

Hypotheses

Ref	Expression
rabeqbii.1	⊢ 𝐴 = 𝐵
rabeqbii.2	⊢ (𝜑 ↔ 𝜓)

Assertion

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

Proof of Theorem rabeqbii

Step	Hyp	Ref	Expression
1		rabeqbii.1	. . . . 5 ⊢ 𝐴 = 𝐵
2	1	eleq2i 2832	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3		rabeqbii.2	. . . 4 ⊢ (𝜑 ↔ 𝜓)
4	2, 3	anbi12i 628	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))
5	4	abbii 2808	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜓)}
6		df-rab 3436	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
7		df-rab 3436	. 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜓)}
8	5, 6, 7	3eqtr4i 2774	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜓}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2713  {crab 3435

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-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436

This theorem is referenced by: (None)