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Theorem mpteq2dv 4109
Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 23-Aug-2014.)
Hypothesis
Ref Expression
mpteq2dv.1 (𝜑𝐵 = 𝐶)
Assertion
Ref Expression
mpteq2dv (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem mpteq2dv
StepHypRef Expression
1 mpteq2dv.1 . . 3 (𝜑𝐵 = 𝐶)
21adantr 276 . 2 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
32mpteq2dva 4108 1 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2160  cmpt 4079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-ral 2473  df-opab 4080  df-mpt 4081
This theorem is referenced by:  ofeqd  6109  ofeq  6110  rdgeq1  6397  rdgeq2  6398  omv  6481  oeiv  6482  0tonninf  10472  1tonninf  10473  iseqf1olemjpcl  10528  iseqf1olemqpcl  10529  iseqf1olemfvp  10530  seq3f1olemqsum  10533  seq3f1olemp  10535  summodc  11426  zsumdc  11427  fsum3  11430  prodeq2w  11599  prodmodc  11621  zproddc  11622  fprodseq  11626  1arithlem1  12398  sloteq  12520  qusex  12805  grplactfval  13060  cnprcl2k  14183  fsumcncntop  14533  expcncf  14569  dvexp  14652  dvexp2  14653  lgsval  14883  peano4nninf  15234  peano3nninf  15235  nninfalllem1  15236  nninfsellemdc  15238  nninfsellemeq  15242  nninfsellemqall  15243  nninfsellemeqinf  15244  nninfomni  15247
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