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Theorem cbvmptg 5137
 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 2379. See cbvmpt 5136 for a version with more disjoint variable conditions, but not requiring ax-13 2379. (Contributed by NM, 11-Sep-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbvmptg.1 𝑦𝐵
cbvmptg.2 𝑥𝐶
cbvmptg.3 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbvmptg (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem cbvmptg
StepHypRef Expression
1 nfcv 2919 . 2 𝑥𝐴
2 nfcv 2919 . 2 𝑦𝐴
3 cbvmptg.1 . 2 𝑦𝐵
4 cbvmptg.2 . 2 𝑥𝐶
5 cbvmptg.3 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
61, 2, 3, 4, 5cbvmptfg 5135 1 (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  Ⅎwnfc 2899   ↦ cmpt 5115 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-v 3411  df-un 3865  df-sn 4526  df-pr 4528  df-op 4532  df-opab 5098  df-mpt 5116 This theorem is referenced by:  cbvmptvg  5139
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