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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 2289. This is the standard proof of the implication in modal logic (B5 ⇒ 4). Its dual statement is bj-modal4e 34993. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-modal4 | ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-modalbe 34966 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑥∃𝑥∀𝑥𝜑) | |
2 | hbe1a 2139 | . 2 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑) | |
3 | 1, 2 | sylg 1824 | 1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1538 ∃wex 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-10 2136 ax-12 2170 |
This theorem depends on definitions: df-bi 206 df-ex 1781 |
This theorem is referenced by: bj-modal4e 34993 bj-substax12 34999 bj-nnfa1 35037 |
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