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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 2291. This is the standard proof of the implication in modal logic (B5 ⇒ 4). Its dual statement is bj-modal4e 36697. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-modal4 | ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-modalbe 36670 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑥∃𝑥∀𝑥𝜑) | |
2 | hbe1a 2141 | . 2 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑) | |
3 | 1, 2 | sylg 1819 | 1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1534 ∃wex 1775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-10 2138 ax-12 2174 |
This theorem depends on definitions: df-bi 207 df-ex 1776 |
This theorem is referenced by: bj-modal4e 36697 bj-substax12 36703 bj-nnfa1 36741 |
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