Theorem cbvrabv2w 42630

Description: A more general version of cbvrabv 3424. Version of cbvrabv2 42629 with a disjoint variable condition, which does not require ax-13 2373. (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 2908	. 2 ⊢ Ⅎ𝑦𝐴
2		nfcv 2908	. 2 ⊢ Ⅎ𝑥𝐵
3		nfv 1920	. 2 ⊢ Ⅎ𝑦𝜑
4		nfv 1920	. 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 3880	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓}

Syntax hints:   ↔ wb 205   = wceq 1541  {crab 3069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837

This theorem is referenced by:  smfsuplem2  44296