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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-modald | Structured version Visualization version GIF version |
Description: A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021.) |
Ref | Expression |
---|---|
bj-modald | ⊢ (∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.2 1981 | . . 3 ⊢ (∀𝑥𝜑 → ∃𝑥𝜑) | |
2 | df-ex 1783 | . . 3 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑) | |
3 | 1, 2 | sylib 217 | . 2 ⊢ (∀𝑥𝜑 → ¬ ∀𝑥 ¬ 𝜑) |
4 | 3 | con2i 139 | 1 ⊢ (∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1540 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-6 1972 |
This theorem depends on definitions: df-bi 206 df-ex 1783 |
This theorem is referenced by: (None) |
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