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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-modalbe | Structured version Visualization version GIF version |
Description: The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2294. (Contributed by BJ, 20-Oct-2019.) |
Ref | Expression |
---|---|
bj-modalbe | ⊢ (𝜑 → ∀𝑥∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modal-b 2294 | . 2 ⊢ (𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑) | |
2 | df-ex 1824 | . . 3 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑) | |
3 | 2 | biimpri 220 | . 2 ⊢ (¬ ∀𝑥 ¬ 𝜑 → ∃𝑥𝜑) |
4 | 1, 3 | sylg 1866 | 1 ⊢ (𝜑 → ∀𝑥∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1599 ∃wex 1823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-10 2135 ax-12 2163 |
This theorem depends on definitions: df-bi 199 df-ex 1824 |
This theorem is referenced by: (None) |
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