Theorem wan 136

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

Assertion

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

Proof of Theorem wan

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

Step	Hyp	Ref	Expression
1		wv 64	. . . . . . 7 |- f:(* -> (* -> *)):(* -> (* -> *))
2		wv 64	. . . . . . 7 |- p:*:*
3		wv 64	. . . . . . 7 |- q:*:*
4	1, 2, 3	wov 72	. . . . . 6 |- [p:*f:(* -> (* -> *))q:*]:*
5	4	wl 66	. . . . 5 |- \f:(* -> (* -> *)) [p:*f:(* -> (* -> *))q:*]:((* -> (* -> *)) -> *)
6		wtru 43	. . . . . . 7 |- T.:*
7	1, 6, 6	wov 72	. . . . . 6 |- [T.f:(* -> (* -> *))T.]:*
8	7	wl 66	. . . . 5 |- \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]:((* -> (* -> *)) -> *)
9	5, 8	weqi 76	. . . 4 |- [\f:(* -> (* -> *)) [p:*f:(* -> (* -> *))q:*] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]]:*
10	9	wl 66	. . 3 |- \q:* [\f:(* -> (* -> *)) [p:*f:(* -> (* -> *))q:*] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]]:(* -> *)
11	10	wl 66	. 2 |- \p:* \q:* [\f:(* -> (* -> *)) [p:*f:(* -> (* -> *))q:*] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]]:(* -> (* -> *))
12		df-an 128	. 2 |- T. |= [ /\ = \p:* \q:* [\f:(* -> (* -> *)) [p:*f:(* -> (* -> *))q:*] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]]]
13	11, 12	eqtypri 81	1 |- /\ :(* -> (* -> *))

Syntax hints:  tv 1   -> ht 2  *hb 3  \kl 6   = ke 7  T.kt 8  [kbr 9  wffMMJ2t 12   /\ tan 119

This theorem was proved from axioms:  ax-cb1 29  ax-weq 40  ax-refl 42  ax-wv 63  ax-wl 65  ax-wov 71  ax-eqtypri 80

This theorem depends on definitions:  df-an 128

This theorem is referenced by:  wim  137  imval  146  anval  148  dfan2  154  hbct  155  ex  158  axrep  220  axun  222