Theorem cbvrabv2w 41468

Description: A more general version of cbvrabv 3488. Version of cbvrabv2 41467 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Revised by Gino Giotto, 16-Apr-2024.)

Hypotheses

Ref	Expression
cbvrabv2w.1	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
cbvrabv2w.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvrabv2w	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓}

Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem cbvrabv2w

Step	Hyp	Ref	Expression
1		nfcv 2976	. 2 ⊢ Ⅎ𝑦𝐴
2		nfcv 2976	. 2 ⊢ Ⅎ𝑥𝐵
3		nfv 1914	. 2 ⊢ Ⅎ𝑦𝜑
4		nfv 1914	. 2 ⊢ Ⅎ𝑥𝜓
5		cbvrabv2w.1	. 2 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
6		cbvrabv2w.2	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
7	1, 2, 3, 4, 5, 6	cbvrabcsfw 3917	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1536  {crab 3141

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877

This theorem is referenced by:  smfsuplem2  43160