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Theorem cbvmptvg 5263
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 2375. See cbvmptv 5261 for a version with more disjoint variable conditions, but not requiring ax-13 2375. (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 2903 . 2 𝑦𝐵
2 nfcv 2903 . 2 𝑥𝐶
3 cbvmptvg.1 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
41, 2, 3cbvmptg 5260 1 (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  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: (None)
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