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Theorem mpt2eq123i 5568
 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 9 . . 3 (⊤ → 𝐴 = 𝐷)
3 mpt2eq123i.2 . . . 4 𝐵 = 𝐸
43a1i 9 . . 3 (⊤ → 𝐵 = 𝐸)
5 mpt2eq123i.3 . . . 4 𝐶 = 𝐹
65a1i 9 . . 3 (⊤ → 𝐶 = 𝐹)
72, 4, 6mpt2eq123dv 5567 . 2 (⊤ → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹))
87trud 1252 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹)
 Colors of variables: wff set class Syntax hints:   = wceq 1243  ⊤wtru 1244   ↦ cmpt2 5514 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-oprab 5516  df-mpt2 5517 This theorem is referenced by:  ofmres  5763
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