notnot1 - Higher-Order Logic Explorer

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Theorem notnot1 150

Description: One side of notnot 187.

**Hypothesis**
Ref	Expression
notval2.1	⊢ A:∗

**Assertion**
Ref	Expression
notnot1	⊢ A⊧(¬ (¬ A))

**Proof of Theorem notnot1**
Step	Hyp	Ref	Expression
1		wfal 125	. . . 4 ⊢ ⊥:∗
2		notval2.1	. . . . 5 ⊢ A:∗
3		wnot 128	. . . . . 6 ⊢ ¬ :(∗ → ∗)
4	3, 2	wc 45	. . . . 5 ⊢ (¬ A):∗
5	2, 4	simpl 22	. . . 4 ⊢ (A, (¬ A))⊧A
6	2, 4	simpr 23	. . . . 5 ⊢ (A, (¬ A))⊧(¬ A)
7	5	ax-cb1 29	. . . . . 6 ⊢ (A, (¬ A)):∗
8	2	notval 135	. . . . . 6 ⊢ ⊤⊧[(¬ A) = [A ⇒ ⊥]]
9	7, 8	a1i 28	. . . . 5 ⊢ (A, (¬ A))⊧[(¬ A) = [A ⇒ ⊥]]
10	6, 9	mpbi 72	. . . 4 ⊢ (A, (¬ A))⊧[A ⇒ ⊥]
11	1, 5, 10	mpd 146	. . 3 ⊢ (A, (¬ A))⊧⊥
12	11	ex 148	. 2 ⊢ A⊧[(¬ A) ⇒ ⊥]
13	4	notval 135	. . 3 ⊢ ⊤⊧[(¬ (¬ A)) = [(¬ A) ⇒ ⊥]]
14	2, 13	a1i 28	. 2 ⊢ A⊧[(¬ (¬ A)) = [(¬ A) ⇒ ⊥]]
15	12, 14	mpbir 77	1 ⊢ A⊧(¬ (¬ A))

Colors of variables: type var term

Syntax hints: ∗hb 3 kc 5 = ke 7 [kbr 9 kct 10 ⊧wffMMJ2 11 wffMMJ2t 12 ⊥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: con3d 152 exnal1 175 notnot 187 ax9 199