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Mirrors > Home > MPE Home > Th. List > Mathboxes > hbntal | Structured version Visualization version GIF version |
Description: A closed form of hbn 2146. hbnt 2144 is another closed form of hbn 2146. (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 2151 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑥(𝜑 → ∀𝑥𝜑)) | |
2 | axc7 2132 | . . . . 5 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑) | |
3 | 2 | con1i 144 | . . . 4 ⊢ (¬ 𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) |
4 | con3 149 | . . . . 5 ⊢ ((𝜑 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝜑)) | |
5 | 4 | al2imi 1743 | . . . 4 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑)) |
6 | 3, 5 | syl5 34 | . . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑)) |
7 | 6 | alimi 1739 | . 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 1481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-12 2047 |
This theorem depends on definitions: df-bi 197 df-or 385 df-ex 1705 df-nf 1710 |
This theorem is referenced by: hbimpg 38770 hbimpgVD 39140 hbexgVD 39142 |
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