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Theorem cbvmptvg 5257
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2377. See cbvmptv 5255 for a version with more disjoint variable conditions, but not requiring ax-13 2377. (Contributed by Mario Carneiro, 19-Feb-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
cbvmptvg.1 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbvmptvg (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbvmptvg
StepHypRef Expression
1 nfcv 2905 . 2 𝑦𝐵
2 nfcv 2905 . 2 𝑥𝐶
3 cbvmptvg.1 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
41, 2, 3cbvmptg 5254 1 (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  cmpt 5225
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2377  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-opab 5206  df-mpt 5226
This theorem is referenced by: (None)
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