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Theorem cbvmpo 7528
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 2904 . 2 𝑧𝐵
2 nfcv 2904 . 2 𝑥𝐵
3 cbvmpo.1 . 2 𝑧𝐶
4 cbvmpo.2 . 2 𝑤𝐶
5 cbvmpo.3 . 2 𝑥𝐷
6 cbvmpo.4 . 2 𝑦𝐷
7 eqidd 2737 . 2 (𝑥 = 𝑧𝐵 = 𝐵)
8 cbvmpo.5 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝐷)
91, 2, 3, 4, 5, 6, 7, 8cbvmpox 7527 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wnfc 2889  cmpo 7434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-opab 5205  df-oprab 7436  df-mpo 7437
This theorem is referenced by:  fvmpopr2d  7596  el2mpocsbcl  8111  fnmpoovd  8113  fmpoco  8121  mpocurryd  8295  fvmpocurryd  8297  xpf1o  9180  cnfcomlem  9740  fseqenlem1  10065  relexpsucnnr  15065  gsumdixp  20317  evlslem4  22101  madugsum  22650  cnmpt2t  23682  cnmptk2  23695  fmucnd  24302  fsum2cn  24896  aks6d1c7lem3  42184  fmpocos  42275  fmuldfeqlem1  45602  smflim  46797
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