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Theorem bj-modal4 36100
Description: First-order logic form of the modal axiom (4). See hba1 2281. This is the standard proof of the implication in modal logic (B5 4). Its dual statement is bj-modal4e 36101. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-modal4 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)

Proof of Theorem bj-modal4
StepHypRef Expression
1 bj-modalbe 36074 . 2 (∀𝑥𝜑 → ∀𝑥𝑥𝑥𝜑)
2 hbe1a 2132 . 2 (∃𝑥𝑥𝜑 → ∀𝑥𝜑)
31, 2sylg 1817 1 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-ex 1774
This theorem is referenced by:  bj-modal4e  36101  bj-substax12  36107  bj-nnfa1  36145
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