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Theorem mpteq2dv 4095
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 4094 1 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  cmpt 4065
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-opab 4066  df-mpt 4067
This theorem is referenced by:  ofeq  6085  rdgeq1  6372  rdgeq2  6373  omv  6456  oeiv  6457  0tonninf  10439  1tonninf  10440  iseqf1olemjpcl  10495  iseqf1olemqpcl  10496  iseqf1olemfvp  10497  seq3f1olemqsum  10500  seq3f1olemp  10502  summodc  11391  zsumdc  11392  fsum3  11395  prodeq2w  11564  prodmodc  11586  zproddc  11587  fprodseq  11591  1arithlem1  12361  sloteq  12467  grplactfval  12971  cnprcl2k  13709  fsumcncntop  14059  expcncf  14095  dvexp  14178  dvexp2  14179  lgsval  14408  peano4nninf  14758  peano3nninf  14759  nninfalllem1  14760  nninfsellemdc  14762  nninfsellemeq  14766  nninfsellemqall  14767  nninfsellemeqinf  14768  nninfomni  14771
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