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Theorem cbvmpt2 5611
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 2194 . 2 𝑧𝐵
2 nfcv 2194 . 2 𝑥𝐵
3 cbvmpt2.1 . 2 𝑧𝐶
4 cbvmpt2.2 . 2 𝑤𝐶
5 cbvmpt2.3 . 2 𝑥𝐷
6 cbvmpt2.4 . 2 𝑦𝐷
7 eqidd 2057 . 2 (𝑥 = 𝑧𝐵 = 𝐵)
8 cbvmpt2.5 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝐷)
91, 2, 3, 4, 5, 6, 7, 8cbvmpt2x 5610 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐵𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wnfc 2181  cmpt2 5542
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847  df-oprab 5544  df-mpt2 5545
This theorem is referenced by:  cbvmpt2v  5612  fmpt2co  5865
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