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Theorem mpteq12i 5212
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 5197 . 2 (⊤ → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
65mptru 1549 1 (𝑥𝐴𝐵) = (𝑥𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wtru 1543  cmpt 5189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-opab 5169  df-mpt 5190
This theorem is referenced by:  partfun  6649  evlsval  21512  madufval  22002  cdj3lem3  31422  cdj3lem3b  31424  esumsnf  32720  esumrnmpt2  32724  measinb2  32879  eulerpart  33039  fiblem  33055  hoidmvlelem4  44925  smflimsup  45155
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