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Theorem cbvmptg 5221
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 5220 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 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 2904 . 2 𝑥𝐴
2 nfcv 2904 . 2 𝑦𝐴
3 cbvmptg.1 . 2 𝑦𝐵
4 cbvmptg.2 . 2 𝑥𝐶
5 cbvmptg.3 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
61, 2, 3, 4, 5cbvmptfg 5219 1 (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wnfc 2884  cmpt 5192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-13 2371  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-opab 5172  df-mpt 5193
This theorem is referenced by:  cbvmptvg  5224
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