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Theorem olc 164
 Description: Or introduction. (Contributed by Mario Carneiro, 9-Oct-2014.)
Hypotheses
Ref Expression
olc.1 A:∗
olc.2 B:∗
Assertion
Ref Expression
olc B⊧[A B]

Proof of Theorem olc
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 wim 137 . . . 4 ⇒ :(∗ → (∗ → ∗))
2 olc.1 . . . . 5 A:∗
3 wv 64 . . . . 5 x:∗:∗
41, 2, 3wov 72 . . . 4 [Ax:∗]:∗
5 olc.2 . . . . . 6 B:∗
61, 5, 3wov 72 . . . . 5 [Bx:∗]:∗
71, 6, 3wov 72 . . . 4 [[Bx:∗] ⇒ x:∗]:∗
81, 4, 7wov 72 . . 3 [[Ax:∗] ⇒ [[Bx:∗] ⇒ x:∗]]:∗
9 wtru 43 . . . 4 ⊤:∗
105, 6simpl 22 . . . . . . . . 9 (B, [Bx:∗])⊧B
115, 6simpr 23 . . . . . . . . 9 (B, [Bx:∗])⊧[Bx:∗]
123, 10, 11mpd 156 . . . . . . . 8 (B, [Bx:∗])⊧x:∗
1312ex 158 . . . . . . 7 B⊧[[Bx:∗] ⇒ x:∗]
1413, 4adantr 55 . . . . . 6 (B, [Ax:∗])⊧[[Bx:∗] ⇒ x:∗]
1514ex 158 . . . . 5 B⊧[[Ax:∗] ⇒ [[Bx:∗] ⇒ x:∗]]
1615eqtru 86 . . . 4 B⊧[⊤ = [[Ax:∗] ⇒ [[Bx:∗] ⇒ x:∗]]]
179, 16eqcomi 79 . . 3 B⊧[[[Ax:∗] ⇒ [[Bx:∗] ⇒ x:∗]] = ⊤]
188, 17leq 91 . 2 B⊧[λx:∗ [[Ax:∗] ⇒ [[Bx:∗] ⇒ x:∗]] = λx:∗ ⊤]
19 wor 140 . . . . 5 :(∗ → (∗ → ∗))
2019, 2, 5wov 72 . . . 4 [A B]:∗
212, 5orval 147 . . . 4 ⊤⊧[[A B] = (λx:∗ [[Ax:∗] ⇒ [[Bx:∗] ⇒ x:∗]])]
228wl 66 . . . . 5 λx:∗ [[Ax:∗] ⇒ [[Bx:∗] ⇒ x:∗]]:(∗ → ∗)
2322alval 142 . . . 4 ⊤⊧[(λx:∗ [[Ax:∗] ⇒ [[Bx:∗] ⇒ x:∗]]) = [λx:∗ [[Ax:∗] ⇒ [[Bx:∗] ⇒ x:∗]] = λx:∗ ⊤]]
2420, 21, 23eqtri 95 . . 3 ⊤⊧[[A B] = [λx:∗ [[Ax:∗] ⇒ [[Bx:∗] ⇒ x:∗]] = λx:∗ ⊤]]
255, 24a1i 28 . 2 B⊧[[A B] = [λx:∗ [[Ax:∗] ⇒ [[Bx:∗] ⇒ x:∗]] = λx:∗ ⊤]]
2618, 25mpbir 87 1 B⊧[A B]
 Colors of variables: type var term Syntax hints:  tv 1  ∗hb 3  kc 5  λkl 6   = ke 7  ⊤kt 8  [kbr 9  kct 10  ⊧wffMMJ2 11  wffMMJ2t 12   ⇒ tim 121  ∀tal 122   ∨ tor 124 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-ded 46  ax-wct 47  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  df-al 126  df-an 128  df-im 129  df-or 132 This theorem is referenced by:  exmid  199
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