Theorem bj-modal4

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

	Ref	Expression
Assertion	bj-modal4	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		bj-modalbe	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑥 ∃ 𝑥 ∀ 𝑥 𝜑 )
2		hbe1a	⊢ ( ∃ 𝑥 ∀ 𝑥 𝜑 → ∀ 𝑥 𝜑 )
3	1 2	sylg	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑥 𝜑 )