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Theorem mpteq12i 3948
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 3942 . 2 (⊤ → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
65mptru 1305 1 (𝑥𝐴𝐵) = (𝑥𝐶𝐷)
Colors of variables: wff set class
Syntax hints:   = wceq 1296  wtru 1297  cmpt 3921
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-11 1449  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-ral 2375  df-opab 3922  df-mpt 3923
This theorem is referenced by:  offres  5944
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