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Theorem wan 136
 Description: Conjunction type. (Contributed by Mario Carneiro, 8-Oct-2014.)
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 64 . . . . . . 7 f:(∗ → (∗ → ∗)):(∗ → (∗ → ∗))
2 wv 64 . . . . . . 7 p:∗:∗
3 wv 64 . . . . . . 7 q:∗:∗
41, 2, 3wov 72 . . . . . 6 [p:∗f:(∗ → (∗ → ∗))q:∗]:∗
54wl 66 . . . . 5 λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗]:((∗ → (∗ → ∗)) → ∗)
6 wtru 43 . . . . . . 7 ⊤:∗
71, 6, 6wov 72 . . . . . 6 [⊤f:(∗ → (∗ → ∗))⊤]:∗
87wl 66 . . . . 5 λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]:((∗ → (∗ → ∗)) → ∗)
95, 8weqi 76 . . . 4 [λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]:∗
109wl 66 . . 3 λq:∗ [λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]:(∗ → ∗)
1110wl 66 . 2 λp:∗ λq:∗ [λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]:(∗ → (∗ → ∗))
12 df-an 128 . 2 ⊤⊧[ = λp:∗ λq:∗ [λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]]
1311, 12eqtypri 81 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 119 This theorem was proved from axioms:  ax-cb1 29  ax-weq 40  ax-refl 42  ax-wv 63  ax-wl 65  ax-wov 71  ax-eqtypri 80 This theorem depends on definitions:  df-an 128 This theorem is referenced by:  wim  137  imval  146  anval  148  dfan2  154  hbct  155  ex  158  axrep  220  axun  222
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