Theorem bj-axc4

Description: Over minimal calculus, the modal axiom (4) ( hba1 ) and the modal axiom (K) ( ax-4 ) together imply axc4 . (Contributed by BJ, 29-Nov-2020) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	bj-axc4	⊢ ( ( ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑥 𝜑 ) → ( ( ∀ 𝑥 ( ∀ 𝑥 𝜑 → 𝜓 ) → ( ∀ 𝑥 ∀ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) → ( ∀ 𝑥 ( ∀ 𝑥 𝜑 → 𝜓 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-imim21	⊢ ( ( ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑥 𝜑 ) → ( ( ∀ 𝑥 ( ∀ 𝑥 𝜑 → 𝜓 ) → ( ∀ 𝑥 ∀ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) → ( ∀ 𝑥 ( ∀ 𝑥 𝜑 → 𝜓 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) ) )