Theorem cbvmptg 5213

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 2371. See cbvmpt 5212 for a version with more disjoint variable conditions, but not requiring ax-13 2371. (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 2892	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2892	. 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 5211	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)

Syntax hints:   → wi 4   = wceq 1540  Ⅎwnfc 2877   ↦ cmpt 5191

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-13 2371  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-opab 5173  df-mpt 5192

This theorem is referenced by:  cbvmptvg  5215