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Theorem wex 129
Description: There exists type.
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 124 . . . 4 |- A.:((* -> *) -> *)
2 wim 127 . . . . . 6 |- ==> :(* -> (* -> *))
3 wal 124 . . . . . . 7 |- A.:((al -> *) -> *)
4 wv 58 . . . . . . . . . 10 |- p:(al -> *):(al -> *)
5 wv 58 . . . . . . . . . 10 |- x:al:al
64, 5wc 45 . . . . . . . . 9 |- (p:(al -> *)x:al):*
7 wv 58 . . . . . . . . 9 |- q:*:*
82, 6, 7wov 64 . . . . . . . 8 |- [(p:(al -> *)x:al) ==> q:*]:*
98wl 59 . . . . . . 7 |- \x:al [(p:(al -> *)x:al) ==> q:*]:(al -> *)
103, 9wc 45 . . . . . 6 |- (A.\x:al [(p:(al -> *)x:al) ==> q:*]):*
112, 10, 7wov 64 . . . . 5 |- [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*]:*
1211wl 59 . . . 4 |- \q:* [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*]:(* -> *)
131, 12wc 45 . . 3 |- (A.\q:* [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*]):*
1413wl 59 . 2 |- \p:(al -> *) (A.\q:* [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*]):((al -> *) -> *)
15 df-ex 121 . 2 |- T. |= [E. = \p:(al -> *) (A.\q:* [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*])]
1614, 15eqtypri 71 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 111  A.tal 112  E.tex 113
This theorem is referenced by:  weu  131  exval  133  euval  134  exlimdv2  156  exlimd  171  eximdv  173  alnex  174  exnal1  175  exnal  188  ax9  199  axrep  207  axun  209
This theorem was proved from axioms:  ax-cb1 29  ax-refl 39
This theorem depends on definitions:  df-al 116  df-an 118  df-im 119  df-ex 121
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