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Theorem ovl 117
 Description: Evaluate a lambda expression in a binary operation. (Contributed by Mario Carneiro, 8-Oct-2014.)
Hypotheses
Ref Expression
ovl.1 A:γ
ovl.2 S:α
ovl.3 T:β
ovl.4 [x:α = S]⊧[A = B]
ovl.5 [y:β = T]⊧[B = C]
Assertion
Ref Expression
ovl ⊤⊧[[Sλx:α λy:β AT] = C]
Distinct variable groups:   x,B   y,C   x,y,S   y,T   α,x   β,y

Proof of Theorem ovl
StepHypRef Expression
1 ovl.1 . . . . 5 A:γ
21wl 66 . . . 4 λy:β A:(βγ)
32wl 66 . . 3 λx:α λy:β A:(α → (βγ))
4 ovl.2 . . 3 S:α
5 ovl.3 . . 3 T:β
63, 4, 5wov 72 . 2 [Sλx:α λy:β AT]:γ
7 weq 41 . . . 4 = :(γ → (γ → ∗))
83, 4wc 50 . . . . 5 (λx:α λy:β AS):(βγ)
98, 5wc 50 . . . 4 ((λx:α λy:β AS)T):γ
10 wtru 43 . . . . 5 ⊤:∗
113, 4, 5df-ov 73 . . . . 5 ⊤⊧(( = [Sλx:α λy:β AT])((λx:α λy:β AS)T))
1210, 11a1i 28 . . . 4 ⊤⊧(( = [Sλx:α λy:β AT])((λx:α λy:β AS)T))
137, 6, 9, 12dfov2 75 . . 3 ⊤⊧[[Sλx:α λy:β AT] = ((λx:α λy:β AS)T)]
141, 4distrl 94 . . . . 5 ⊤⊧[(λx:α λy:β AS) = λy:β (λx:α AS)]
1510, 14a1i 28 . . . 4 ⊤⊧[(λx:α λy:β AS) = λy:β (λx:α AS)]
168, 5, 15ceq1 89 . . 3 ⊤⊧[((λx:α λy:β AS)T) = (λy:β (λx:α AS)T)]
176, 13, 16eqtri 95 . 2 ⊤⊧[[Sλx:α λy:β AT] = (λy:β (λx:α AS)T)]
181wl 66 . . . 4 λx:α A:(αγ)
1918, 4wc 50 . . 3 (λx:α AS):γ
20 wv 64 . . . . . 6 y:β:β
2120, 5weqi 76 . . . . 5 [y:β = T]:∗
22 ovl.4 . . . . . 6 [x:α = S]⊧[A = B]
231, 4, 22cl 116 . . . . 5 ⊤⊧[(λx:α AS) = B]
2421, 23a1i 28 . . . 4 [y:β = T]⊧[(λx:α AS) = B]
25 ovl.5 . . . 4 [y:β = T]⊧[B = C]
2619, 24, 25eqtri 95 . . 3 [y:β = T]⊧[(λx:α AS) = C]
2719, 5, 26cl 116 . 2 ⊤⊧[(λy:β (λx:α AS)T) = C]
286, 17, 27eqtri 95 1 ⊤⊧[[Sλx:α λy:β AT] = C]
 Colors of variables: type var term Syntax hints:  tv 1   → ht 2  kc 5  λkl 6   = ke 7  ⊤kt 8  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12 This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-simpl 20  ax-simpr 21  ax-id 24  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-wctl 31  ax-wctr 32  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-distrl 70  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113 This theorem depends on definitions:  df-ov 73 This theorem is referenced by:  imval  146  orval  147  anval  148  dfan2  154
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