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Theorem mpt2eq123i 5664
 Description: An equality inference for the maps to notation. (Contributed by set.mm contributors, 15-Jul-2013.)
Hypotheses
Ref Expression
mpt2eq123i.1 A = D
mpt2eq123i.2 B = E
mpt2eq123i.3 C = F
Assertion
Ref Expression
mpt2eq123i (x A, y B C) = (x D, y E F)

Proof of Theorem mpt2eq123i
StepHypRef Expression
1 mpt2eq123i.1 . . . 4 A = D
21a1i 10 . . 3 ( ⊤ → A = D)
3 mpt2eq123i.2 . . . 4 B = E
43a1i 10 . . 3 ( ⊤ → B = E)
5 mpt2eq123i.3 . . . 4 C = F
65a1i 10 . . 3 ( ⊤ → C = F)
72, 4, 6mpt2eq123dv 5663 . 2 ( ⊤ → (x A, y B C) = (x D, y E F))
87trud 1323 1 (x A, y B C) = (x D, y E F)
 Colors of variables: wff setvar class Syntax hints:   ⊤ wtru 1316   = wceq 1642   ↦ cmpt2 5653 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-oprab 5528  df-mpt2 5654 This theorem is referenced by: (None)
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