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Theorem cbvmpov 5898
 Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4059, 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 2299 . 2 𝑧𝐶
2 nfcv 2299 . 2 𝑤𝐶
3 nfcv 2299 . 2 𝑥𝐷
4 nfcv 2299 . 2 𝑦𝐷
5 cbvmpov.1 . . 3 (𝑥 = 𝑧𝐶 = 𝐸)
6 cbvmpov.2 . . 3 (𝑦 = 𝑤𝐸 = 𝐷)
75, 6sylan9eq 2210 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝐷)
81, 2, 3, 4, 7cbvmpo 5897 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐵𝐷)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1335   ∈ cmpo 5823 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-opab 4026  df-oprab 5825  df-mpo 5826 This theorem is referenced by:  frec2uzrdg  10301  frecuzrdgsuc  10306  iseqvalcbv  10349  resqrexlemfp1  10902  resqrex  10919  sqne2sq  12042  ennnfonelemnn0  12134  txbas  12629  xmetxp  12878
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