Theorem cbvrabvOLD 3489

Description: Obsolete version of cbvrabv 3488 as of 14-Jun-2023. Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
cbvrabvOLD.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

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

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

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

Proof of Theorem cbvrabvOLD

Step	Hyp	Ref	Expression
1		nfcv 2976	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2976	. 2 ⊢ Ⅎ𝑦𝐴
3		nfv 1914	. 2 ⊢ Ⅎ𝑦𝜑
4		nfv 1914	. 2 ⊢ Ⅎ𝑥𝜓
5		cbvrabvOLD.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	cbvrab 3487	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 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-13 2389  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  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

This theorem is referenced by: (None)