MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvmpt2 Structured version   Visualization version   GIF version

Theorem cbvmpt2 6687
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by NM, 17-Dec-2013.)
Hypotheses
Ref Expression
cbvmpt2.1 𝑧𝐶
cbvmpt2.2 𝑤𝐶
cbvmpt2.3 𝑥𝐷
cbvmpt2.4 𝑦𝐷
cbvmpt2.5 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝐷)
Assertion
Ref Expression
cbvmpt2 (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 cbvmpt2
StepHypRef Expression
1 nfcv 2761 . 2 𝑧𝐵
2 nfcv 2761 . 2 𝑥𝐵
3 cbvmpt2.1 . 2 𝑧𝐶
4 cbvmpt2.2 . 2 𝑤𝐶
5 cbvmpt2.3 . 2 𝑥𝐷
6 cbvmpt2.4 . 2 𝑦𝐷
7 eqidd 2622 . 2 (𝑥 = 𝑧𝐵 = 𝐵)
8 cbvmpt2.5 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝐷)
91, 2, 3, 4, 5, 6, 7, 8cbvmpt2x 6686 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wnfc 2748  cmpt2 6606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-opab 4674  df-oprab 6608  df-mpt2 6609
This theorem is referenced by:  cbvmpt2v  6688  el2mpt2csbcl  7195  fnmpt2ovd  7197  fmpt2co  7205  mpt2curryd  7340  fvmpt2curryd  7342  xpf1o  8066  cnfcomlem  8540  fseqenlem1  8791  relexpsucnnr  13699  gsumdixp  18530  evlslem4  19427  madugsum  20368  cnmpt2t  21386  cnmptk2  21399  fmucnd  22006  fsum2cn  22582  fmuldfeqlem1  39215  smflim  40289
  Copyright terms: Public domain W3C validator