Theorem cbvmptg 5215

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 2404. See cbvmpt 5214 for a version with more disjoint variable conditions, but not requiring ax-13 2404. (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 2925	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2925	. 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 5213	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)

Syntax hints:   → wi 4   = wceq 1570  Ⅎwnfc 2910   ↦ cmpt 5193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-13 2404  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-opab 5175  df-mpt 5194

This theorem is referenced by:  cbvmptvg  5217