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Theorem mpt2eq123i 6616
 Description: An equality inference for the maps to notation. (Contributed by NM, 15-Jul-2013.)
Hypotheses
Ref Expression
mpt2eq123i.1 𝐴 = 𝐷
mpt2eq123i.2 𝐵 = 𝐸
mpt2eq123i.3 𝐶 = 𝐹
Assertion
Ref Expression
mpt2eq123i (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹)

Proof of Theorem mpt2eq123i
StepHypRef Expression
1 mpt2eq123i.1 . . . 4 𝐴 = 𝐷
21a1i 11 . . 3 (⊤ → 𝐴 = 𝐷)
3 mpt2eq123i.2 . . . 4 𝐵 = 𝐸
43a1i 11 . . 3 (⊤ → 𝐵 = 𝐸)
5 mpt2eq123i.3 . . . 4 𝐶 = 𝐹
65a1i 11 . . 3 (⊤ → 𝐶 = 𝐹)
72, 4, 6mpt2eq123dv 6615 . 2 (⊤ → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹))
87trud 1484 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  ⊤wtru 1476   ↦ cmpt2 6551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  ofmres  7055  seqval  12674  oppgtmd  21711  mdetlap1  29220  sdc  32710  tgrpset  35051  mendvscafval  36779  fsovcnvlem  37327  hspmbl  39519  wlkson  40864
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