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Theorem cbvmpo 7503
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 7502 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wnfc 2884  cmpo 7411
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 5300  ax-nul 5307  ax-pr 5428
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 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5212  df-oprab 7413  df-mpo 7414
This theorem is referenced by:  cbvmpov  7504  fvmpopr2d  7569  el2mpocsbcl  8071  fnmpoovd  8073  fmpoco  8081  mpocurryd  8254  fvmpocurryd  8256  xpf1o  9139  cnfcomlem  9694  fseqenlem1  10019  relexpsucnnr  14972  gsumdixp  20131  evlslem4  21637  madugsum  22145  cnmpt2t  23177  cnmptk2  23190  fmucnd  23797  fsum2cn  24387  fmpocos  41056  fmuldfeqlem1  44298  smflim  45493
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