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Mirrors > Home > MPE Home > Th. List > Mathboxes > hbntg | Structured version Visualization version GIF version |
Description: A more general form of hbnt 2293. (Contributed by Scott Fenton, 13-Dec-2010.) |
Ref | Expression |
---|---|
hbntg | ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axc7 2327 | . . 3 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜓 → 𝜓) | |
2 | 1 | con1i 149 | . 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ ∀𝑥𝜓) |
3 | con3 156 | . . 3 ⊢ ((𝜑 → ∀𝑥𝜓) → (¬ ∀𝑥𝜓 → ¬ 𝜑)) | |
4 | 3 | al2imi 1807 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → (∀𝑥 ¬ ∀𝑥𝜓 → ∀𝑥 ¬ 𝜑)) |
5 | 2, 4 | syl5 34 | 1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-ex 1772 |
This theorem is referenced by: hbimtg 32948 hbng 32950 |
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