Theorem notval2 149

Description: Another way two write ~ A, the negation of A.

Hypothesis

Ref	Expression
notval2.1	|- A:*

Assertion

Ref	Expression
notval2	|- T. |= [(~ A) = [A = F.]]

Proof of Theorem notval2

Step	Hyp	Ref	Expression
1		wnot 128	. . 3 |- ~ :(* -> *)
2		notval2.1	. . 3 |- A:*
3	1, 2	wc 45	. 2 |- (~ A):*
4	2	notval 135	. 2 |- T. |= [(~ A) = [A ==> F.]]
5		wfal 125	. . . . 5 |- F.:*
6		wim 127	. . . . . . 7 |- ==> :(* -> (* -> *))
7	6, 2, 5	wov 64	. . . . . 6 |- [A ==> F.]:*
8	7, 2	simpr 23	. . . . 5 |- ([A ==> F.], A) |= A
9	7, 2	simpl 22	. . . . 5 |- ([A ==> F.], A) |= [A ==> F.]
10	5, 8, 9	mpd 146	. . . 4 |- ([A ==> F.], A) |= F.
11	2	pm2.21 143	. . . . 5 |- F. |= A
12	11, 7	adantl 51	. . . 4 |- ([A ==> F.], F.) |= A
13	10, 12	ded 74	. . 3 |- [A ==> F.] |= [A = F.]
14	13	ax-cb2 30	. . . . . 6 |- [A = F.]:*
15	14, 2	simpr 23	. . . . 5 |- ([A = F.], A) |= A
16	14, 2	simpl 22	. . . . 5 |- ([A = F.], A) |= [A = F.]
17	15, 16	mpbi 72	. . . 4 |- ([A = F.], A) |= F.
18	17	ex 148	. . 3 |- [A = F.] |= [A ==> F.]
19	13, 18	dedi 75	. 2 |- T. |= [[A ==> F.] = [A = F.]]
20	3, 4, 19	eqtri 85	1 |- T. |= [(~ A) = [A = F.]]

Syntax hints:  *hb 3  kc 5   = ke 7  T.kt 8  [kbr 9  kct 10   |= wffMMJ2 11  wffMMJ2t 12  F.tfal 108  ~ tne 110   ==> tim 111

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-refl 39  ax-eqmp 42  ax-ded 43  ax-ceq 46  ax-beta 60  ax-distrc 61  ax-leq 62  ax-distrl 63  ax-hbl1 93  ax-17 95  ax-inst 103

This theorem depends on definitions:  df-ov 65  df-al 116  df-fal 117  df-an 118  df-im 119  df-not 120

This theorem is referenced by: (None)