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Theorem mpteq12i 5254
Description: An equality inference for the maps-to notation. (Contributed by Scott Fenton, 27-Oct-2010.) (Revised by Mario Carneiro, 16-Dec-2013.)
Hypotheses
Ref Expression
mpteq12i.1 𝐴 = 𝐶
mpteq12i.2 𝐵 = 𝐷
Assertion
Ref Expression
mpteq12i (𝑥𝐴𝐵) = (𝑥𝐶𝐷)

Proof of Theorem mpteq12i
StepHypRef Expression
1 mpteq12i.1 . . . 4 𝐴 = 𝐶
21a1i 11 . . 3 (⊤ → 𝐴 = 𝐶)
3 mpteq12i.2 . . . 4 𝐵 = 𝐷
43a1i 11 . . 3 (⊤ → 𝐵 = 𝐷)
52, 4mpteq12dv 5239 . 2 (⊤ → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
65mptru 1547 1 (𝑥𝐴𝐵) = (𝑥𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wtru 1541  cmpt 5231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-opab 5211  df-mpt 5232
This theorem is referenced by:  partfun  6697  evlsval  21869  madufval  22360  cdj3lem3  31959  cdj3lem3b  31961  esumsnf  33361  esumrnmpt2  33365  measinb2  33520  eulerpart  33680  fiblem  33696  hoidmvlelem4  45613  smflimsup  45843
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