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Mirrors > Home > MPE Home > Th. List > Mathboxes > hbntal | Structured version Visualization version GIF version |
Description: A closed form of hbn 2289. hbnt 2288 is another closed form of hbn 2289. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hbntal | ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hba1 2287 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑥(𝜑 → ∀𝑥𝜑)) | |
2 | axc7 2308 | . . . . 5 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑) | |
3 | 2 | con1i 147 | . . . 4 ⊢ (¬ 𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) |
4 | con3 153 | . . . . 5 ⊢ ((𝜑 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝜑)) | |
5 | 4 | al2imi 1815 | . . . 4 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑)) |
6 | 3, 5 | syl5 34 | . . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑)) |
7 | 6 | alimi 1811 | . 2 ⊢ (∀𝑥∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑)) |
8 | 1, 7 | syl 17 | 1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-10 2135 ax-12 2169 |
This theorem depends on definitions: df-bi 206 df-or 844 df-ex 1780 df-nf 1784 |
This theorem is referenced by: hbimpg 43619 hbimpgVD 43969 hbexgVD 43971 |
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