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Theorem mpteq12i 5155
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 5147 . 2 (⊤ → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
65mptru 1537 1 (𝑥𝐴𝐵) = (𝑥𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  wtru 1531  cmpt 5142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-12 2167  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-opab 5125  df-mpt 5143
This theorem is referenced by:  evlsval  20219  madufval  21162  cdj3lem3  30130  cdj3lem3b  30132  partfun  30337  esumsnf  31210  esumrnmpt2  31214  measinb2  31369  eulerpart  31527  fiblem  31543  hoidmvlelem4  42743  smflimsup  42965
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