Theorem notval 145

Description: Value of negation. (Contributed by Mario Carneiro, 8-Oct-2014.)

Hypothesis

Ref	Expression
imval.1	|- A:*

Assertion

Ref	Expression
notval	|- T. |= [(~ A) = [A ==> F.]]

Proof of Theorem notval

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wnot 138	. . 3 |- ~ :(* -> *)
2		imval.1	. . 3 |- A:*
3	1, 2	wc 50	. 2 |- (~ A):*
4		df-not 130	. . 3 |- T. |= [~ = \p:* [p:* ==> F.]]
5	1, 2, 4	ceq1 89	. 2 |- T. |= [(~ A) = (\p:* [p:* ==> F.]A)]
6		wim 137	. . . 4 |- ==> :(* -> (* -> *))
7		wv 64	. . . 4 |- p:*:*
8		wfal 135	. . . 4 |- F.:*
9	6, 7, 8	wov 72	. . 3 |- [p:* ==> F.]:*
10	7, 2	weqi 76	. . . . 5 |- [p:* = A]:*
11	10	id 25	. . . 4 |- [p:* = A] |= [p:* = A]
12	6, 7, 8, 11	oveq1 99	. . 3 |- [p:* = A] |= [[p:* ==> F.] = [A ==> F.]]
13	9, 2, 12	cl 116	. 2 |- T. |= [(\p:* [p:* ==> F.]A) = [A ==> F.]]
14	3, 5, 13	eqtri 95	1 |- T. |= [(~ A) = [A ==> F.]]

Syntax hints:  tv 1  *hb 3  kc 5  \kl 6   = ke 7  T.kt 8  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12  F.tfal 118  ~ tne 120   ==> tim 121

This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-simpl 20  ax-simpr 21  ax-id 24  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-wctl 31  ax-wctr 32  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113

This theorem depends on definitions:  df-ov 73  df-al 126  df-fal 127  df-an 128  df-im 129  df-not 130

This theorem is referenced by:  notval2  159  notnot1  160  con2d  161  alnex  186  exmid  199  notnot  200  ax3  205