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.

Step	Hyp	Ref	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
6	4, 5	wc 45	. . . . . . . . 9 |- (p:(al -> *)x:al):*
7		wv 58	. . . . . . . . 9 |- q:*:*
8	2, 6, 7	wov 64	. . . . . . . 8 |- [(p:(al -> *)x:al) ==> q:*]:*
9	8	wl 59	. . . . . . 7 |- \x:al [(p:(al -> *)x:al) ==> q:*]:(al -> *)
10	3, 9	wc 45	. . . . . 6 |- (A.\x:al [(p:(al -> *)x:al) ==> q:*]):*
11	2, 10, 7	wov 64	. . . . 5 |- [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*]:*
12	11	wl 59	. . . 4 |- \q:* [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*]:(* -> *)
13	1, 12	wc 45	. . 3 |- (A.\q:* [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*]):*
14	13	wl 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:*])]
16	14, 15	eqtypri 71	1 |- E.:((al -> *) -> *)

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 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

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