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Theorem cbvmpo 5951
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by NM, 17-Dec-2013.)
Hypotheses
Ref Expression
cbvmpo.1 𝑧𝐶
cbvmpo.2 𝑤𝐶
cbvmpo.3 𝑥𝐷
cbvmpo.4 𝑦𝐷
cbvmpo.5 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝐷)
Assertion
Ref Expression
cbvmpo (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐵𝐷)
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝐵,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧,𝑤)   𝐷(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cbvmpo
StepHypRef Expression
1 nfcv 2319 . 2 𝑧𝐵
2 nfcv 2319 . 2 𝑥𝐵
3 cbvmpo.1 . 2 𝑧𝐶
4 cbvmpo.2 . 2 𝑤𝐶
5 cbvmpo.3 . 2 𝑥𝐷
6 cbvmpo.4 . 2 𝑦𝐷
7 eqidd 2178 . 2 (𝑥 = 𝑧𝐵 = 𝐵)
8 cbvmpo.5 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝐷)
91, 2, 3, 4, 5, 6, 7, 8cbvmpox 5950 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐵𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wnfc 2306  cmpo 5874
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 4120  ax-pow 4173  ax-pr 4208
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 4064  df-oprab 5876  df-mpo 5877
This theorem is referenced by:  cbvmpov  5952  fnmpoovd  6213  fmpoco  6214  xpf1o  6841  cnmpt2t  13664
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