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Theorem wex 139
Description: There exists type. (Contributed by Mario Carneiro, 8-Oct-2014.)
Assertion
Ref Expression
wex |- E.:((al -> *) -> *)

Proof of Theorem wex
Dummy variables p q x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wal 134 . . . 4 |- A.:((* -> *) -> *)
2 wim 137 . . . . . 6 |- ==> :(* -> (* -> *))
3 wal 134 . . . . . . 7 |- A.:((al -> *) -> *)
4 wv 64 . . . . . . . . . 10 |- p:(al -> *):(al -> *)
5 wv 64 . . . . . . . . . 10 |- x:al:al
64, 5wc 50 . . . . . . . . 9 |- (p:(al -> *)x:al):*
7 wv 64 . . . . . . . . 9 |- q:*:*
82, 6, 7wov 72 . . . . . . . 8 |- [(p:(al -> *)x:al) ==> q:*]:*
98wl 66 . . . . . . 7 |- \x:al [(p:(al -> *)x:al) ==> q:*]:(al -> *)
103, 9wc 50 . . . . . 6 |- (A.\x:al [(p:(al -> *)x:al) ==> q:*]):*
112, 10, 7wov 72 . . . . 5 |- [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*]:*
1211wl 66 . . . 4 |- \q:* [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*]:(* -> *)
131, 12wc 50 . . 3 |- (A.\q:* [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*]):*
1413wl 66 . 2 |- \p:(al -> *) (A.\q:* [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*]):((al -> *) -> *)
15 df-ex 131 . 2 |- T. |= [E. = \p:(al -> *) (A.\q:* [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*])]
1614, 15eqtypri 81 1 |- E.:((al -> *) -> *)
Colors of variables: type var term
Syntax hints:  tv 1   -> ht 2  *hb 3  kc 5  \kl 6  T.kt 8  [kbr 9  wffMMJ2t 12   ==> tim 121  A.tal 122  E.tex 123
This theorem was proved from axioms:  ax-cb1 29  ax-weq 40  ax-refl 42  ax-wc 49  ax-wv 63  ax-wl 65  ax-wov 71  ax-eqtypri 80
This theorem depends on definitions:  df-al 126  df-an 128  df-im 129  df-ex 131
This theorem is referenced by:  weu  141  exval  143  euval  144  exlimdv2  166  exlimd  183  eximdv  185  alnex  186  exnal1  187  exnal  201  ax9  212  axrep  220  axun  222
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