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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 |- \f:(* -> (* -> *)) [p:*f:(* -> (* -> *))q:*]:((* -> (* -> *)) -> *)
6 wtru 43 . . . . . . 7 |- T.:*
71, 6, 6wov 72 . . . . . 6 |- [T.f:(* -> (* -> *))T.]:*
87wl 66 . . . . 5 |- \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]:((* -> (* -> *)) -> *)
95, 8weqi 76 . . . 4 |- [\f:(* -> (* -> *)) [p:*f:(* -> (* -> *))q:*] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]]:*
109wl 66 . . 3 |- \q:* [\f:(* -> (* -> *)) [p:*f:(* -> (* -> *))q:*] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]]:(* -> *)
1110wl 66 . 2 |- \p:* \q:* [\f:(* -> (* -> *)) [p:*f:(* -> (* -> *))q:*] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]]:(* -> (* -> *))
12 df-an 128 . 2 |- T. |= [ /\ = \p:* \q:* [\f:(* -> (* -> *)) [p:*f:(* -> (* -> *))q:*] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]]]
1311, 12eqtypri 81 1 |- /\ :(* -> (* -> *))
Colors of variables: type var term
Syntax hints:  tv 1   -> ht 2  *hb 3  \kl 6   = ke 7  T.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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