Theorem wor 140

Description: Disjunction type. (Contributed by Mario Carneiro, 8-Oct-2014.)

Assertion

Ref	Expression
wor	|- \/ :(* -> (* -> *))

Proof of Theorem wor

Dummy variables p q x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wal 134	. . . . 5 |- A.:((* -> *) -> *)
2		wim 137	. . . . . . 7 |- ==> :(* -> (* -> *))
3		wv 64	. . . . . . . 8 |- p:*:*
4		wv 64	. . . . . . . 8 |- x:*:*
5	2, 3, 4	wov 72	. . . . . . 7 |- [p:* ==> x:*]:*
6		wv 64	. . . . . . . . 9 |- q:*:*
7	2, 6, 4	wov 72	. . . . . . . 8 |- [q:* ==> x:*]:*
8	2, 7, 4	wov 72	. . . . . . 7 |- [[q:* ==> x:*] ==> x:*]:*
9	2, 5, 8	wov 72	. . . . . 6 |- [[p:* ==> x:*] ==> [[q:* ==> x:*] ==> x:*]]:*
10	9	wl 66	. . . . 5 |- \x:* [[p:* ==> x:*] ==> [[q:* ==> x:*] ==> x:*]]:(* -> *)
11	1, 10	wc 50	. . . 4 |- (A.\x:* [[p:* ==> x:*] ==> [[q:* ==> x:*] ==> x:*]]):*
12	11	wl 66	. . 3 |- \q:* (A.\x:* [[p:* ==> x:*] ==> [[q:* ==> x:*] ==> x:*]]):(* -> *)
13	12	wl 66	. 2 |- \p:* \q:* (A.\x:* [[p:* ==> x:*] ==> [[q:* ==> x:*] ==> x:*]]):(* -> (* -> *))
14		df-or 132	. 2 |- T. |= [ \/ = \p:* \q:* (A.\x:* [[p:* ==> x:*] ==> [[q:* ==> x:*] ==> x:*]])]
15	13, 14	eqtypri 81	1 |- \/ :(* -> (* -> *))

Syntax hints:  tv 1   -> ht 2  *hb 3  kc 5  \kl 6  T.kt 8  [kbr 9  wffMMJ2t 12   ==> tim 121  A.tal 122   \/ tor 124

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-or 132

This theorem is referenced by:  orval  147  olc  164  orc  165  exmid  199