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Theorem cbvmpov 5954
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4098, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)
Hypotheses
Ref Expression
cbvmpov.1 (𝑥 = 𝑧𝐶 = 𝐸)
cbvmpov.2 (𝑦 = 𝑤𝐸 = 𝐷)
Assertion
Ref Expression
cbvmpov (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐵𝐷)
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝐵,𝑥,𝑦,𝑧   𝑤,𝐶,𝑧   𝑥,𝐷,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑧,𝑤)   𝐸(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cbvmpov
StepHypRef Expression
1 nfcv 2319 . 2 𝑧𝐶
2 nfcv 2319 . 2 𝑤𝐶
3 nfcv 2319 . 2 𝑥𝐷
4 nfcv 2319 . 2 𝑦𝐷
5 cbvmpov.1 . . 3 (𝑥 = 𝑧𝐶 = 𝐸)
6 cbvmpov.2 . . 3 (𝑦 = 𝑤𝐸 = 𝐷)
75, 6sylan9eq 2230 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝐷)
81, 2, 3, 4, 7cbvmpo 5953 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐵𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  cmpo 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-opab 4065  df-oprab 5878  df-mpo 5879
This theorem is referenced by:  frec2uzrdg  10408  frecuzrdgsuc  10413  iseqvalcbv  10456  resqrexlemfp1  11017  resqrex  11034  sqne2sq  12176  ennnfonelemnn0  12422  nninfdc  12453  txbas  13728  xmetxp  13977
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