Theorem cbvmptg 5182

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2372. See cbvmpt 5181 for a version with more disjoint variable conditions, but not requiring ax-13 2372. (Contributed by NM, 11-Sep-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cbvmptg.1	⊢ Ⅎ𝑦𝐵
cbvmptg.2	⊢ Ⅎ𝑥𝐶
cbvmptg.3	⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

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

Proof of Theorem cbvmptg

Step	Hyp	Ref	Expression
1		nfcv 2906	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2906	. 2 ⊢ Ⅎ𝑦𝐴
3		cbvmptg.1	. 2 ⊢ Ⅎ𝑦𝐵
4		cbvmptg.2	. 2 ⊢ Ⅎ𝑥𝐶
5		cbvmptg.3	. 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
6	1, 2, 3, 4, 5	cbvmptfg 5180	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)

Syntax hints:   → wi 4   = wceq 1539  Ⅎwnfc 2886   ↦ cmpt 5153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-13 2372  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-opab 5133  df-mpt 5154

This theorem is referenced by:  cbvmptvg  5185