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Theorem cbvmpt 5166
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.) Add disjoint variable condition to avoid ax-13 2386. See cbvmptg 5167 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.)
Hypotheses
Ref Expression
cbvmpt.1 𝑦𝐵
cbvmpt.2 𝑥𝐶
cbvmpt.3 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbvmpt (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem cbvmpt
StepHypRef Expression
1 nfcv 2977 . 2 𝑥𝐴
2 nfcv 2977 . 2 𝑦𝐴
3 cbvmpt.1 . 2 𝑦𝐵
4 cbvmpt.2 . 2 𝑥𝐶
5 cbvmpt.3 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
61, 2, 3, 4, 5cbvmptf 5164 1 (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wnfc 2961  cmpt 5145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-opab 5128  df-mpt 5146
This theorem is referenced by:  cbvmptv  5168  dffn5f  6735  fvmpts  6770  fvmpt2i  6777  fvmptex  6781  fmptcof  6891  fmptcos  6892  fliftfuns  7066  offval2  7425  ofmpteq  7427  mpocurryvald  7935  qliftfuns  8383  axcc2  9858  seqof2  13427  summolem2a  15071  zsum  15074  fsumcvg2  15083  fsumrlim  15165  cbvprod  15268  prodmolem2a  15287  zprod  15290  fprod  15294  pcmptdvds  16229  prdsdsval2  16756  gsumconstf  19054  gsummpt1n0  19084  gsum2d2  19093  dprd2d2  19165  gsumdixp  19358  psrass1lem  20156  coe1fzgsumdlem  20468  gsumply1eq  20472  evl1gsumdlem  20518  madugsum  21251  cnmpt1t  22272  cnmpt2k  22295  elmptrab  22434  flfcnp2  22614  prdsxmet  22978  fsumcn  23477  ovoliunlem3  24104  ovoliun  24105  ovoliun2  24106  voliun  24154  mbfpos  24251  mbfposb  24253  i1fposd  24307  itg2cnlem1  24361  isibl2  24366  cbvitg  24375  itgss3  24414  itgfsum  24426  itgabs  24434  itgcn  24442  limcmpt  24480  dvmptfsum  24571  lhop2  24611  dvfsumle  24617  dvfsumlem2  24623  itgsubstlem  24644  itgsubst  24645  itgulm2  24996  rlimcnp2  25543  gsummpt2co  30686  esumsnf  31323  mbfposadd  34938  itgabsnc  34960  ftc1cnnclem  34964  ftc2nc  34975  mzpsubst  39345  rabdiophlem2  39399  aomclem8  39661  fsumcnf  41278  disjf1  41443  disjrnmpt2  41449  disjinfi  41454  fmptf  41509  cncfmptss  41868  mulc1cncfg  41870  expcnfg  41872  fprodcn  41881  fnlimabslt  41960  climmptf  41962  liminfvalxr  42064  liminfpnfuz  42097  xlimpnfxnegmnf2  42139  icccncfext  42170  cncficcgt0  42171  cncfiooicclem1  42176  fprodcncf  42184  dvmptmulf  42222  iblsplitf  42255  stoweidlem21  42307  stirlinglem4  42363  stirlinglem13  42372  stirlinglem15  42374  fourierd  42508  fourierclimd  42509  sge0iunmptlemre  42698  sge0iunmpt  42701  sge0ltfirpmpt2  42709  sge0isummpt2  42715  sge0xaddlem2  42717  sge0xadd  42718  meadjiun  42749  meaiunincf  42766  meaiuninc3  42768  omeiunle  42800  caratheodorylem2  42810  ovncvrrp  42847  vonioo  42965  smflim2  43081  smfsup  43089  smfinf  43093  smflimsup  43103  smfliminf  43106
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