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Theorem mpteq2dv 4094
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 4093 1 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  cmpt 4064
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 4065  df-mpt 4066
This theorem is referenced by:  ofeq  6084  rdgeq1  6371  rdgeq2  6372  omv  6455  oeiv  6456  0tonninf  10438  1tonninf  10439  iseqf1olemjpcl  10494  iseqf1olemqpcl  10495  iseqf1olemfvp  10496  seq3f1olemqsum  10499  seq3f1olemp  10501  summodc  11390  zsumdc  11391  fsum3  11394  prodeq2w  11563  prodmodc  11585  zproddc  11586  fprodseq  11590  1arithlem1  12360  sloteq  12466  grplactfval  12970  cnprcl2k  13676  fsumcncntop  14026  expcncf  14062  dvexp  14145  dvexp2  14146  lgsval  14375  peano4nninf  14725  peano3nninf  14726  nninfalllem1  14727  nninfsellemdc  14729  nninfsellemeq  14733  nninfsellemqall  14734  nninfsellemeqinf  14735  nninfomni  14738
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