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Theorem mpteq12i 4023
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 9 . . 3 (⊤ → 𝐴 = 𝐶)
3 mpteq12i.2 . . . 4 𝐵 = 𝐷
43a1i 9 . . 3 (⊤ → 𝐵 = 𝐷)
52, 4mpteq12dv 4017 . 2 (⊤ → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
65mptru 1341 1 (𝑥𝐴𝐵) = (𝑥𝐶𝐷)
Colors of variables: wff set class
Syntax hints:   = wceq 1332  wtru 1333  cmpt 3996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-ral 2422  df-opab 3997  df-mpt 3998
This theorem is referenced by:  offres  6040
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