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.

Step	Hyp	Ref	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
6	4, 5	wc 50	. . . . . . . . 9 |- (p:(al -> *)x:al):*
7		wv 64	. . . . . . . . 9 |- q:*:*
8	2, 6, 7	wov 72	. . . . . . . 8 |- [(p:(al -> *)x:al) ==> q:*]:*
9	8	wl 66	. . . . . . 7 |- \x:al [(p:(al -> *)x:al) ==> q:*]:(al -> *)
10	3, 9	wc 50	. . . . . 6 |- (A.\x:al [(p:(al -> *)x:al) ==> q:*]):*
11	2, 10, 7	wov 72	. . . . 5 |- [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*]:*
12	11	wl 66	. . . 4 |- \q:* [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*]:(* -> *)
13	1, 12	wc 50	. . 3 |- (A.\q:* [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*]):*
14	13	wl 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:*])]
16	14, 15	eqtypri 81	1 |- E.:((al -> *) -> *)

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