Theorem cbvrabv2 39625

Description: A more general version of cbvrabv 3230. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
cbvrabv2.1	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
cbvrabv2.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvrabv2	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓}

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

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

Proof of Theorem cbvrabv2

Step	Hyp	Ref	Expression
1		nfcv 2793	. 2 ⊢ Ⅎ𝑦𝐴
2		nfcv 2793	. 2 ⊢ Ⅎ𝑥𝐵
3		nfv 1883	. 2 ⊢ Ⅎ𝑦𝜑
4		nfv 1883	. 2 ⊢ Ⅎ𝑥𝜓
5		cbvrabv2.1	. 2 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
6		cbvrabv2.2	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
7	1, 2, 3, 4, 5, 6	cbvrabcsf 3601	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1523  {crab 2945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-sbc 3469  df-csb 3567

This theorem is referenced by:  smfsuplem2  41339