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Theorem cbvmptv 3880
 Description: Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.)
Hypothesis
Ref Expression
cbvmptv.1 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbvmptv (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbvmptv
StepHypRef Expression
1 nfcv 2194 . 2 𝑦𝐵
2 nfcv 2194 . 2 𝑥𝐶
3 cbvmptv.1 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
41, 2, 3cbvmpt 3879 1 (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1259   ↦ cmpt 3846 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-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 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-sn 3409  df-pr 3410  df-op 3412  df-opab 3847  df-mpt 3848 This theorem is referenced by:  frecsuc  6022  caucvgsrlembnd  6943  frec2uzzd  9350  frec2uzsucd  9351  climcvg1n  10100
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