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Theorem mpteq2dv 3904
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 270 . 2 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
32mpteq2dva 3903 1 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wcel 1436  cmpt 3874
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-ral 2360  df-opab 3875  df-mpt 3876
This theorem is referenced by:  ofeq  5815  rdgeq1  6090  rdgeq2  6091  omv  6170  oeiv  6171  0tonninf  9773  1tonninf  9774  iseqf1olemjpcl  9829  iseqf1olemqpcl  9830  iseqf1olemfvp  9831  iseqf1olemqsum  9834  iseqf1olemp  9836  isummo  10664  zisum  10665  fisum  10667  peano4nninf  11341  peano3nninf  11342  nninfalllem1  11344  nninfsellemdc  11347  nninfsellemeq  11351  nninfsellemqall  11352  nninfsellemeqinf  11353  nninfomni  11356
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