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Theorem cbvmpo 7455
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 2734 . 2 (𝑥 = 𝑧𝐵 = 𝐵)
8 cbvmpo.5 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝐷)
91, 2, 3, 4, 5, 6, 7, 8cbvmpox 7454 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wnfc 2884  cmpo 7363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-opab 5172  df-oprab 7365  df-mpo 7366
This theorem is referenced by:  cbvmpov  7456  fvmpopr2d  7520  el2mpocsbcl  8021  fnmpoovd  8023  fmpoco  8031  mpocurryd  8204  fvmpocurryd  8206  xpf1o  9089  cnfcomlem  9643  fseqenlem1  9968  relexpsucnnr  14919  gsumdixp  20041  evlslem4  21507  madugsum  22015  cnmpt2t  23047  cnmptk2  23060  fmucnd  23667  fsum2cn  24257  fmuldfeqlem1  43913  smflim  45108
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