exnal1 - Higher-Order Logic Explorer

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Theorem exnal1 175

Description: Forward direction of exnal 188.

**Hypothesis**
Ref	Expression
alnex1.1	⊢ A:∗

**Assertion**
Ref	Expression
exnal1	⊢ (∃λx:α (¬ A))⊧(¬ (∀λx:α A))

Distinct variable group: α,x

**Proof of Theorem exnal1**
Step	Hyp	Ref	Expression
1		wex 129	. . . 4 ⊢ ∃:((α → ∗) → ∗)
2		wnot 128	. . . . . 6 ⊢ ¬ :(∗ → ∗)
3		alnex1.1	. . . . . 6 ⊢ A:∗
4	2, 3	wc 45	. . . . 5 ⊢ (¬ A):∗
5	4	wl 59	. . . 4 ⊢ λx:α (¬ A):(α → ∗)
6	1, 5	wc 45	. . 3 ⊢ (∃λx:α (¬ A)):∗
7	3	notnot1 150	. . . . . 6 ⊢ A⊧(¬ (¬ A))
8		wtru 40	. . . . . 6 ⊢ ⊤:∗
9	7, 8	adantl 51	. . . . 5 ⊢ (⊤, A)⊧(¬ (¬ A))
10	9	alimdv 172	. . . 4 ⊢ (⊤, (∀λx:α A))⊧(∀λx:α (¬ (¬ A)))
11		wal 124	. . . . . . 7 ⊢ ∀:((α → ∗) → ∗)
12	2, 4	wc 45	. . . . . . . 8 ⊢ (¬ (¬ A)):∗
13	12	wl 59	. . . . . . 7 ⊢ λx:α (¬ (¬ A)):(α → ∗)
14	11, 13	wc 45	. . . . . 6 ⊢ (∀λx:α (¬ (¬ A))):∗
15	14	id 25	. . . . 5 ⊢ (∀λx:α (¬ (¬ A)))⊧(∀λx:α (¬ (¬ A)))
16	4	alnex 174	. . . . . 6 ⊢ ⊤⊧[(∀λx:α (¬ (¬ A))) = (¬ (∃λx:α (¬ A)))]
17	14, 16	a1i 28	. . . . 5 ⊢ (∀λx:α (¬ (¬ A)))⊧[(∀λx:α (¬ (¬ A))) = (¬ (∃λx:α (¬ A)))]
18	15, 17	mpbi 72	. . . 4 ⊢ (∀λx:α (¬ (¬ A)))⊧(¬ (∃λx:α (¬ A)))
19	10, 18	syl 16	. . 3 ⊢ (⊤, (∀λx:α A))⊧(¬ (∃λx:α (¬ A)))
20	6, 19	con2d 151	. 2 ⊢ (⊤, (∃λx:α (¬ A)))⊧(¬ (∀λx:α A))
21	20	trul 37	1 ⊢ (∃λx:α (¬ A))⊧(¬ (∀λx:α A))

Colors of variables: type var term

Syntax hints: → ht 2 ∗hb 3 kc 5 λkl 6 = ke 7 ⊤kt 8 [kbr 9 kct 10 ⊧wffMMJ2 11 wffMMJ2t 12 ¬ tne 110 ∀tal 112 ∃tex 113

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 ax-eta 165

This theorem depends on definitions: df-ov 65 df-al 116 df-fal 117 df-an 118 df-im 119 df-not 120 df-ex 121

This theorem is referenced by: (None)