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Theorem mpt2eq123i 6703
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 6702 . 2 (⊤ → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹))
87trud 1491 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1481  wtru 1482  cmpt2 6637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-oprab 6639  df-mpt2 6640
This theorem is referenced by:  ofmres  7149  seqval  12795  oppgtmd  21882  wlkson  26533  mdetlap1  29866  sdc  33511  tgrpset  35852  mendvscafval  37579  fsovcnvlem  38127  hspmbl  40606
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