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	⊢ ⊤⊧[(¬ A) = [A = ⊥]]

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 ⊢ ⊤⊧[(¬ A) = [A ⇒ ⊥]]
5		wfal 135	. . . . 5 ⊢ ⊥:∗
6		wim 137	. . . . . . 7 ⊢ ⇒ :(∗ → (∗ → ∗))
7	6, 2, 5	wov 72	. . . . . 6 ⊢ [A ⇒ ⊥]:∗
8	7, 2	simpr 23	. . . . 5 ⊢ ([A ⇒ ⊥], A)⊧A
9	7, 2	simpl 22	. . . . 5 ⊢ ([A ⇒ ⊥], A)⊧[A ⇒ ⊥]
10	5, 8, 9	mpd 156	. . . 4 ⊢ ([A ⇒ ⊥], A)⊧⊥
11	2	pm2.21 153	. . . . 5 ⊢ ⊥⊧A
12	11, 7	adantl 56	. . . 4 ⊢ ([A ⇒ ⊥], ⊥)⊧A
13	10, 12	ded 84	. . 3 ⊢ [A ⇒ ⊥]⊧[A = ⊥]
14	13	ax-cb2 30	. . . . . 6 ⊢ [A = ⊥]:∗
15	14, 2	simpr 23	. . . . 5 ⊢ ([A = ⊥], A)⊧A
16	14, 2	simpl 22	. . . . 5 ⊢ ([A = ⊥], A)⊧[A = ⊥]
17	15, 16	mpbi 82	. . . 4 ⊢ ([A = ⊥], A)⊧⊥
18	17	ex 158	. . 3 ⊢ [A = ⊥]⊧[A ⇒ ⊥]
19	13, 18	dedi 85	. 2 ⊢ ⊤⊧[[A ⇒ ⊥] = [A = ⊥]]
20	3, 4, 19	eqtri 95	1 ⊢ ⊤⊧[(¬ A) = [A = ⊥]]

Syntax hints:  ∗hb 3  kc 5   = ke 7  ⊤kt 8  [kbr 9  kct 10  ⊧wffMMJ2 11  wffMMJ2t 12  ⊥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)