Theorem bj-modal4 34069

Description: First-order logic form of the modal axiom (4). See hba1 2300. This is the standard proof of the implication in modal logic (B5 ⇒ 4). Its dual statement is bj-modal4e 34070. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-modal4	⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

Proof of Theorem bj-modal4

Step	Hyp	Ref	Expression
1		bj-modalbe 34043	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥∃𝑥∀𝑥𝜑)
2		hbe1a 2147	. 2 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)
3	1, 2	sylg 1822	1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1534  ∃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 1969  ax-7 2014  ax-10 2144  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-ex 1780

This theorem is referenced by:  bj-modal4e  34070  bj-nnfa1  34109