Theorem cbvmptvOLD 5256

Description: Obsolete version of cbvmptv 5255 as of 17-Nov-2024. (Contributed by Mario Carneiro, 19-Feb-2013.) Add disjoint variable condition to avoid ax-13 2377. See cbvmptvg 5257 for a less restrictive version requiring more axioms. (Revised by GG, 17-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
cbvmptvOLD.1	⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)

Assertion

Ref	Expression
cbvmptvOLD	⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐶

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbvmptvOLD

Step	Hyp	Ref	Expression
1		nfcv 2905	. 2 ⊢ Ⅎ𝑦𝐵
2		nfcv 2905	. 2 ⊢ Ⅎ𝑥𝐶
3		cbvmptvOLD.1	. 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
4	1, 2, 3	cbvmpt 5253	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)

Syntax hints:   → wi 4   = wceq 1540   ↦ cmpt 5225

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-opab 5206  df-mpt 5226

This theorem is referenced by: (None)