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Theorem cbvmpt2v 5636
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 3892, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)
Hypotheses
Ref Expression
cbvmpt2v.1 (𝑥 = 𝑧𝐶 = 𝐸)
cbvmpt2v.2 (𝑦 = 𝑤𝐸 = 𝐷)
Assertion
Ref Expression
cbvmpt2v (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐵𝐷)
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝐵,𝑥,𝑦,𝑧   𝑤,𝐶,𝑧   𝑥,𝐷,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑧,𝑤)   𝐸(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cbvmpt2v
StepHypRef Expression
1 nfcv 2223 . 2 𝑧𝐶
2 nfcv 2223 . 2 𝑤𝐶
3 nfcv 2223 . 2 𝑥𝐷
4 nfcv 2223 . 2 𝑦𝐷
5 cbvmpt2v.1 . . 3 (𝑥 = 𝑧𝐶 = 𝐸)
6 cbvmpt2v.2 . . 3 (𝑦 = 𝑤𝐸 = 𝐷)
75, 6sylan9eq 2135 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝐷)
81, 2, 3, 4, 7cbvmpt2 5635 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐵𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  cmpt2 5566
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-opab 3860  df-oprab 5568  df-mpt2 5569
This theorem is referenced by:  frec2uzrdg  9561  frecuzrdgsuc  9566  iseqvalcbv  9601  resqrexlemfp1  10114  resqrex  10131  sqne2sq  10780
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