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Theorem cbvmptv 3934
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 2228 . 2 𝑦𝐵
2 nfcv 2228 . 2 𝑥𝐶
3 cbvmptv.1 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
41, 2, 3cbvmpt 3933 1 (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  cmpt 3899
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-opab 3900  df-mpt 3901
This theorem is referenced by:  frecsuc  6172  xpmapen  6564  fodjuomni  6802  caucvgsrlembnd  7344  negiso  8414  infrenegsupex  9080  frec2uzsucd  9804  frecuzrdgdom  9821  frecuzrdgfun  9823  frecuzrdgsuct  9827  0tonninf  9841  1tonninf  9842  seq3f1oleml  9928  seq3f1o  9929  hashfz1  10187  climcvg1n  10735  isummo  10769  zisum  10770  fisum  10774  fsumadd  10796  phimullem  11475  nninfsellemqall  11862  nninfomni  11866  exmidsbthrlem  11867  exmidsbthr  11868
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