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Mirrors > Home > HOLE Home > Th. List > ax6 | GIF version |
Description: Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. |
Ref | Expression |
---|---|
ax6.1 | ⊢ R:∗ |
Ref | Expression |
---|---|
ax6 | ⊢ ⊤⊧[(¬ (∀λx:α R)) ⇒ (∀λx:α (¬ (∀λx:α R)))] |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wnot 128 | . . 3 ⊢ ¬ :(∗ → ∗) | |
2 | wal 124 | . . . 4 ⊢ ∀:((α → ∗) → ∗) | |
3 | ax6.1 | . . . . 5 ⊢ R:∗ | |
4 | 3 | wl 59 | . . . 4 ⊢ λx:α R:(α → ∗) |
5 | 2, 4 | wc 45 | . . 3 ⊢ (∀λx:α R):∗ |
6 | 1, 5 | wc 45 | . 2 ⊢ (¬ (∀λx:α R)):∗ |
7 | wv 58 | . . 3 ⊢ y:α:α | |
8 | 1, 7 | ax-17 95 | . . 3 ⊢ ⊤⊧[(λx:α ¬ y:α) = ¬ ] |
9 | 2, 7 | ax-17 95 | . . . 4 ⊢ ⊤⊧[(λx:α ∀y:α) = ∀] |
10 | 3, 7 | ax-hbl1 93 | . . . 4 ⊢ ⊤⊧[(λx:α λx:α Ry:α) = λx:α R] |
11 | 2, 4, 7, 9, 10 | hbc 100 | . . 3 ⊢ ⊤⊧[(λx:α (∀λx:α R)y:α) = (∀λx:α R)] |
12 | 1, 5, 7, 8, 11 | hbc 100 | . 2 ⊢ ⊤⊧[(λx:α (¬ (∀λx:α R))y:α) = (¬ (∀λx:α R))] |
13 | 6, 12 | isfree 176 | 1 ⊢ ⊤⊧[(¬ (∀λx:α R)) ⇒ (∀λx:α (¬ (∀λx:α R)))] |
Colors of variables: type var term |
Syntax hints: tv 1 → ht 2 ∗hb 3 kc 5 λkl 6 ⊤kt 8 [kbr 9 ⊧wffMMJ2 11 wffMMJ2t 12 ¬ tne 110 ⇒ tim 111 ∀tal 112 |
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 |
This theorem is referenced by: (None) |
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