Theorem cbvmptvOLD 5188

Description: Obsolete version of cbvmptv 5187 as of 17-Nov-2024. (Contributed by Mario Carneiro, 19-Feb-2013.) Add disjoint variable condition to avoid ax-13 2372. See cbvmptvg 5189 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 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 2907	. 2 ⊢ Ⅎ𝑦𝐵
2		nfcv 2907	. 2 ⊢ Ⅎ𝑥𝐶
3		cbvmptvOLD.1	. 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
4	1, 2, 3	cbvmpt 5185	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)

Syntax hints:   → wi 4   = wceq 1539   ↦ cmpt 5157

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5137  df-mpt 5158

This theorem is referenced by: (None)