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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-modal4 | Structured version Visualization version GIF version | ||
| Description: First-order logic form of the modal axiom (4). See hba1 2293. This is the standard proof of the implication in modal logic (B5 ⇒ 4). Its dual statement is bj-modal4e 36716. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| bj-modal4 | ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bj-modalbe 36689 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑥∃𝑥∀𝑥𝜑) | |
| 2 | hbe1a 2144 | . 2 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑) | |
| 3 | 1, 2 | sylg 1823 | 1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 ∃wex 1779 |
| 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 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-ex 1780 |
| This theorem is referenced by: bj-modal4e 36716 bj-substax12 36722 bj-nnfa1 36760 |
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