Theorem cbvmptg 5260

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 2375. See cbvmpt 5259 for a version with more disjoint variable conditions, but not requiring ax-13 2375. (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 2903	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2903	. 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 5258	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)

Syntax hints:   → wi 4   = wceq 1537  Ⅎwnfc 2888   ↦ cmpt 5231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-mpt 5232

This theorem is referenced by:  cbvmptvg  5263