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Theorem wan 126
Description: Conjunction type.
Assertion
Ref Expression
wan :(∗ → (∗ → ∗))

Proof of Theorem wan
Dummy variables f p q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wv 58 . . . . . . 7 f:(∗ → (∗ → ∗)):(∗ → (∗ → ∗))
2 wv 58 . . . . . . 7 p:∗:∗
3 wv 58 . . . . . . 7 q:∗:∗
41, 2, 3wov 64 . . . . . 6 [p:∗f:(∗ → (∗ → ∗))q:∗]:∗
54wl 59 . . . . 5 λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗]:((∗ → (∗ → ∗)) → ∗)
6 wtru 40 . . . . . . 7 ⊤:∗
71, 6, 6wov 64 . . . . . 6 [⊤f:(∗ → (∗ → ∗))⊤]:∗
87wl 59 . . . . 5 λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]:((∗ → (∗ → ∗)) → ∗)
95, 8weqi 68 . . . 4 [λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]:∗
109wl 59 . . 3 λq:∗ [λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]:(∗ → ∗)
1110wl 59 . 2 λp:∗ λq:∗ [λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]:(∗ → (∗ → ∗))
12 df-an 118 . 2 ⊤⊧[ = λp:∗ λq:∗ [λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]]
1311, 12eqtypri 71 1 :(∗ → (∗ → ∗))
Colors of variables: type var term
Syntax hints:  tv 1  ht 2  hb 3  λkl 6   = ke 7  kt 8  [kbr 9  wffMMJ2t 12   tan 109
This theorem was proved from axioms:  ax-cb1 29  ax-refl 39
This theorem depends on definitions:  df-an 118
This theorem is referenced by:  wim  127  imval  136  anval  138  dfan2  144  hbct  145  ex  148  axrep  207  axun  209
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