Theorem cbvrabv2 43816

Description: A more general version of cbvrabv 3443. Usage of this theorem is discouraged because it depends on ax-13 2372. Use of cbvrabv2w 43817 is preferred. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (New usage is discouraged.)

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 2904	. 2 ⊢ Ⅎ𝑦𝐴
2		nfcv 2904	. 2 ⊢ Ⅎ𝑥𝐵
3		nfv 1918	. 2 ⊢ Ⅎ𝑦𝜑
4		nfv 1918	. 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 3942	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓}

Syntax hints:   ↔ wb 205   = wceq 1542  {crab 3433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-13 2372  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3434  df-sbc 3779  df-csb 3895

This theorem is referenced by: (None)