Theorem wor 130

Description: Disjunction type.

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 124	. . . . 5 |- A.:((* -> *) -> *)
2		wim 127	. . . . . . 7 |- ==> :(* -> (* -> *))
3		wv 58	. . . . . . . 8 |- p:*:*
4		wv 58	. . . . . . . 8 |- x:*:*
5	2, 3, 4	wov 64	. . . . . . 7 |- [p:* ==> x:*]:*
6		wv 58	. . . . . . . . 9 |- q:*:*
7	2, 6, 4	wov 64	. . . . . . . 8 |- [q:* ==> x:*]:*
8	2, 7, 4	wov 64	. . . . . . 7 |- [[q:* ==> x:*] ==> x:*]:*
9	2, 5, 8	wov 64	. . . . . 6 |- [[p:* ==> x:*] ==> [[q:* ==> x:*] ==> x:*]]:*
10	9	wl 59	. . . . 5 |- \x:* [[p:* ==> x:*] ==> [[q:* ==> x:*] ==> x:*]]:(* -> *)
11	1, 10	wc 45	. . . 4 |- (A.\x:* [[p:* ==> x:*] ==> [[q:* ==> x:*] ==> x:*]]):*
12	11	wl 59	. . 3 |- \q:* (A.\x:* [[p:* ==> x:*] ==> [[q:* ==> x:*] ==> x:*]]):(* -> *)
13	12	wl 59	. 2 |- \p:* \q:* (A.\x:* [[p:* ==> x:*] ==> [[q:* ==> x:*] ==> x:*]]):(* -> (* -> *))
14		df-or 122	. 2 |- T. |= [ \/ = \p:* \q:* (A.\x:* [[p:* ==> x:*] ==> [[q:* ==> x:*] ==> x:*]])]
15	13, 14	eqtypri 71	1 |- \/ :(* -> (* -> *))

Syntax hints:  tv 1   -> ht 2  *hb 3  kc 5  \kl 6  T.kt 8  [kbr 9  wffMMJ2t 12   ==> tim 111  A.tal 112   \/ tor 114

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-or 122

This theorem is referenced by:  orval  137  olc  154  orc  155  exmid  186