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Theorem ceq12 88
 Description: Equality theorem for combination. (Contributed by Mario Carneiro, 7-Oct-2014.)
Hypotheses
Ref Expression
ceq12.1 F:(αβ)
ceq12.2 A:α
ceq12.3 R⊧[F = T]
ceq12.4 R⊧[A = B]
Assertion
Ref Expression
ceq12 R⊧[(FA) = (TB)]

Proof of Theorem ceq12
StepHypRef Expression
1 weq 41 . 2 = :(β → (β → ∗))
2 ceq12.1 . . 3 F:(αβ)
3 ceq12.2 . . 3 A:α
42, 3wc 50 . 2 (FA):β
5 ceq12.3 . . . 4 R⊧[F = T]
62, 5eqtypi 78 . . 3 T:(αβ)
7 ceq12.4 . . . 4 R⊧[A = B]
83, 7eqtypi 78 . . 3 B:α
96, 8wc 50 . 2 (TB):β
10 weq 41 . . . 4 = :((αβ) → ((αβ) → ∗))
1110, 2, 6, 5dfov1 74 . . 3 R⊧(( = F)T)
12 weq 41 . . . 4 = :(α → (α → ∗))
1312, 3, 8, 7dfov1 74 . . 3 R⊧(( = A)B)
142, 6, 3, 8ax-ceq 51 . . 3 ((( = F)T), (( = A)B))⊧(( = (FA))(TB))
1511, 13, 14syl2anc 19 . 2 R⊧(( = (FA))(TB))
161, 4, 9, 15dfov2 75 1 R⊧[(FA) = (TB)]
 Colors of variables: type var term Syntax hints:   → ht 2  kc 5   = ke 7  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12 This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-wc 49  ax-ceq 51  ax-wov 71  ax-eqtypi 77 This theorem depends on definitions:  df-ov 73 This theorem is referenced by:  ceq1  89  ceq2  90  oveq123  98  hbc  110  ac  197
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