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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 |- \f:(* -> (* -> *)) [p:*f:(* -> (* -> *))q:*]:((* -> (* -> *)) -> *)
6 wtru 40 . . . . . . 7 |- T.:*
71, 6, 6wov 64 . . . . . 6 |- [T.f:(* -> (* -> *))T.]:*
87wl 59 . . . . 5 |- \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]:((* -> (* -> *)) -> *)
95, 8weqi 68 . . . 4 |- [\f:(* -> (* -> *)) [p:*f:(* -> (* -> *))q:*] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]]:*
109wl 59 . . 3 |- \q:* [\f:(* -> (* -> *)) [p:*f:(* -> (* -> *))q:*] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]]:(* -> *)
1110wl 59 . 2 |- \p:* \q:* [\f:(* -> (* -> *)) [p:*f:(* -> (* -> *))q:*] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]]:(* -> (* -> *))
12 df-an 118 . 2 |- T. |= [ /\ = \p:* \q:* [\f:(* -> (* -> *)) [p:*f:(* -> (* -> *))q:*] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]]]
1311, 12eqtypri 71 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 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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