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Theorem oveq123 98
 Description: Equality theorem for binary operation. (Contributed by Mario Carneiro, 7-Oct-2014.)
Hypotheses
Ref Expression
oveq.1 F:(α → (βγ))
oveq.2 A:α
oveq.3 B:β
oveq123.4 R⊧[F = S]
oveq123.5 R⊧[A = C]
oveq123.6 R⊧[B = T]
Assertion
Ref Expression
oveq123 R⊧[[AFB] = [CST]]

Proof of Theorem oveq123
StepHypRef Expression
1 oveq.1 . . . 4 F:(α → (βγ))
2 oveq.2 . . . 4 A:α
31, 2wc 50 . . 3 (FA):(βγ)
4 oveq.3 . . 3 B:β
53, 4wc 50 . 2 ((FA)B):γ
6 oveq123.4 . . . 4 R⊧[F = S]
7 oveq123.5 . . . 4 R⊧[A = C]
81, 2, 6, 7ceq12 88 . . 3 R⊧[(FA) = (SC)]
9 oveq123.6 . . 3 R⊧[B = T]
103, 4, 8, 9ceq12 88 . 2 R⊧[((FA)B) = ((SC)T)]
11 weq 41 . . 3 = :(γ → (γ → ∗))
121, 2, 4wov 72 . . 3 [AFB]:γ
136ax-cb1 29 . . . 4 R:∗
141, 2, 4df-ov 73 . . . 4 ⊤⊧(( = [AFB])((FA)B))
1513, 14a1i 28 . . 3 R⊧(( = [AFB])((FA)B))
1611, 12, 5, 15dfov2 75 . 2 R⊧[[AFB] = ((FA)B)]
171, 6eqtypi 78 . . . 4 S:(α → (βγ))
182, 7eqtypi 78 . . . 4 C:α
194, 9eqtypi 78 . . . 4 T:β
2017, 18, 19wov 72 . . 3 [CST]:γ
2117, 18wc 50 . . . 4 (SC):(βγ)
2221, 19wc 50 . . 3 ((SC)T):γ
2317, 18, 19df-ov 73 . . . 4 ⊤⊧(( = [CST])((SC)T))
2413, 23a1i 28 . . 3 R⊧(( = [CST])((SC)T))
2511, 20, 22, 24dfov2 75 . 2 R⊧[[CST] = ((SC)T)]
265, 10, 16, 253eqtr4i 96 1 R⊧[[AFB] = [CST]]
 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  ax-eqtypri 80 This theorem depends on definitions:  df-ov 73 This theorem is referenced by:  oveq1  99  oveq12  100  oveq  102
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