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Theorem anval 148
 Description: Value of the conjunction. (Contributed by Mario Carneiro, 9-Oct-2014.)
Hypotheses
Ref Expression
imval.1 A:∗
imval.2 B:∗
Assertion
Ref Expression
anval ⊤⊧[[A B] = [λf:(∗ → (∗ → ∗)) [Af:(∗ → (∗ → ∗))B] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]]
Distinct variable groups:   f,A   f,B

Proof of Theorem anval
Dummy variables p q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wan 136 . . 3 :(∗ → (∗ → ∗))
2 imval.1 . . 3 A:∗
3 imval.2 . . 3 B:∗
41, 2, 3wov 72 . 2 [A B]:∗
5 df-an 128 . . 3 ⊤⊧[ = λp:∗ λq:∗ [λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]]
61, 2, 3, 5oveq 102 . 2 ⊤⊧[[A B] = [Aλp:∗ λq:∗ [λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]B]]
7 wv 64 . . . . . 6 f:(∗ → (∗ → ∗)):(∗ → (∗ → ∗))
8 wv 64 . . . . . 6 p:∗:∗
9 wv 64 . . . . . 6 q:∗:∗
107, 8, 9wov 72 . . . . 5 [p:∗f:(∗ → (∗ → ∗))q:∗]:∗
1110wl 66 . . . 4 λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗]:((∗ → (∗ → ∗)) → ∗)
12 wtru 43 . . . . . 6 ⊤:∗
137, 12, 12wov 72 . . . . 5 [⊤f:(∗ → (∗ → ∗))⊤]:∗
1413wl 66 . . . 4 λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]:((∗ → (∗ → ∗)) → ∗)
1511, 14weqi 76 . . 3 [λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]:∗
16 weq 41 . . . 4 = :(((∗ → (∗ → ∗)) → ∗) → (((∗ → (∗ → ∗)) → ∗) → ∗))
178, 2weqi 76 . . . . . . 7 [p:∗ = A]:∗
1817id 25 . . . . . 6 [p:∗ = A]⊧[p:∗ = A]
197, 8, 9, 18oveq1 99 . . . . 5 [p:∗ = A]⊧[[p:∗f:(∗ → (∗ → ∗))q:∗] = [Af:(∗ → (∗ → ∗))q:∗]]
2010, 19leq 91 . . . 4 [p:∗ = A]⊧[λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [Af:(∗ → (∗ → ∗))q:∗]]
2116, 11, 14, 20oveq1 99 . . 3 [p:∗ = A]⊧[[λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]] = [λf:(∗ → (∗ → ∗)) [Af:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]]
227, 2, 9wov 72 . . . . 5 [Af:(∗ → (∗ → ∗))q:∗]:∗
2322wl 66 . . . 4 λf:(∗ → (∗ → ∗)) [Af:(∗ → (∗ → ∗))q:∗]:((∗ → (∗ → ∗)) → ∗)
249, 3weqi 76 . . . . . . 7 [q:∗ = B]:∗
2524id 25 . . . . . 6 [q:∗ = B]⊧[q:∗ = B]
267, 2, 9, 25oveq2 101 . . . . 5 [q:∗ = B]⊧[[Af:(∗ → (∗ → ∗))q:∗] = [Af:(∗ → (∗ → ∗))B]]
2722, 26leq 91 . . . 4 [q:∗ = B]⊧[λf:(∗ → (∗ → ∗)) [Af:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [Af:(∗ → (∗ → ∗))B]]
2816, 23, 14, 27oveq1 99 . . 3 [q:∗ = B]⊧[[λf:(∗ → (∗ → ∗)) [Af:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]] = [λf:(∗ → (∗ → ∗)) [Af:(∗ → (∗ → ∗))B] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]]
2915, 2, 3, 21, 28ovl 117 . 2 ⊤⊧[[Aλp:∗ λq:∗ [λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]B] = [λf:(∗ → (∗ → ∗)) [Af:(∗ → (∗ → ∗))B] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]]
304, 6, 29eqtri 95 1 ⊤⊧[[A B] = [λf:(∗ → (∗ → ∗)) [Af:(∗ → (∗ → ∗))B] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]]
 Colors of variables: type var term Syntax hints:  tv 1   → ht 2  ∗hb 3  λkl 6   = ke 7  ⊤kt 8  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12   ∧ tan 119 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  df-an 128 This theorem is referenced by:  dfan2  154
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