MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvmpo Structured version   Visualization version   GIF version

Theorem cbvmpo 7231
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 2958 . 2 𝑧𝐵
2 nfcv 2958 . 2 𝑥𝐵
3 cbvmpo.1 . 2 𝑧𝐶
4 cbvmpo.2 . 2 𝑤𝐶
5 cbvmpo.3 . 2 𝑥𝐷
6 cbvmpo.4 . 2 𝑦𝐷
7 eqidd 2802 . 2 (𝑥 = 𝑧𝐵 = 𝐵)
8 cbvmpo.5 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝐷)
91, 2, 3, 4, 5, 6, 7, 8cbvmpox 7230 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wnfc 2939  cmpo 7141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-opab 5096  df-oprab 7143  df-mpo 7144
This theorem is referenced by:  cbvmpov  7232  fvmpopr2d  7294  el2mpocsbcl  7767  fnmpoovd  7769  fmpoco  7777  mpocurryd  7922  fvmpocurryd  7924  xpf1o  8667  cnfcomlem  9150  fseqenlem1  9439  relexpsucnnr  14380  gsumdixp  19359  evlslem4  20751  madugsum  21252  cnmpt2t  22282  cnmptk2  22295  fmucnd  22902  fsum2cn  23480  fmuldfeqlem1  42221  smflim  43407
  Copyright terms: Public domain W3C validator