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Theorem cbvmptv 3984
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 2255 . 2 𝑦𝐵
2 nfcv 2255 . 2 𝑥𝐶
3 cbvmptv.1 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
41, 2, 3cbvmpt 3983 1 (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  cmpt 3949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-sn 3499  df-pr 3500  df-op 3502  df-opab 3950  df-mpt 3951
This theorem is referenced by:  frecsuc  6258  xpmapen  6697  omp1eom  6932  fodjuomni  6971  fodjumkv  6984  caucvgsrlembnd  7540  negiso  8620  infrenegsupex  9288  frec2uzsucd  10064  frecuzrdgdom  10081  frecuzrdgfun  10083  frecuzrdgsuct  10087  0tonninf  10102  1tonninf  10103  seq3f1oleml  10166  seq3f1o  10167  hashfz1  10419  xrnegiso  10920  infxrnegsupex  10921  climcvg1n  11008  summodc  11041  zsumdc  11042  fsum3  11045  fsumadd  11064  phimullem  11743  ennnfonelemnn0  11777  ennnfonelemr  11778  ctinfom  11783  cdivcncfap  12570  expcncf  12575  subctctexmid  12880  nninfsellemqall  12895  nninfomni  12899  nninffeq  12900  exmidsbthrlem  12901  exmidsbthr  12902  isomninn  12910
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