Theorem notval2 159

Description: Another way two write ~ A, the negation of A. (Contributed by Mario Carneiro, 9-Oct-2014.)

Hypothesis

Ref	Expression
notval2.1	|- A:*

Assertion

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

Proof of Theorem notval2

Step	Hyp	Ref	Expression
1		wnot 138	. . 3 |- ~ :(* -> *)
2		notval2.1	. . 3 |- A:*
3	1, 2	wc 50	. 2 |- (~ A):*
4	2	notval 145	. 2 |- T. |= [(~ A) = [A ==> F.]]
5		wfal 135	. . . . 5 |- F.:*
6		wim 137	. . . . . . 7 |- ==> :(* -> (* -> *))
7	6, 2, 5	wov 72	. . . . . 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 156	. . . 4 |- ([A ==> F.], A) |= F.
11	2	pm2.21 153	. . . . 5 |- F. |= A
12	11, 7	adantl 56	. . . 4 |- ([A ==> F.], F.) |= A
13	10, 12	ded 84	. . 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 82	. . . 4 |- ([A = F.], A) |= F.
18	17	ex 158	. . 3 |- [A = F.] |= [A ==> F.]
19	13, 18	dedi 85	. 2 |- T. |= [[A ==> F.] = [A = F.]]
20	3, 4, 19	eqtri 95	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 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-ded 46  ax-wct 47  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-distrl 70  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: (None)