Theorem bj-alrim

Description: Closed form of alrimi . (Contributed by BJ, 2-May-2019)

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

Proof

Step	Hyp	Ref	Expression
1		nf5r	⊢ ( Ⅎ 𝑥 𝜑 → ( 𝜑 → ∀ 𝑥 𝜑 ) )
2		sylgt	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ( 𝜑 → ∀ 𝑥 𝜑 ) → ( 𝜑 → ∀ 𝑥 𝜓 ) ) )
3	1 2	syl5com	⊢ ( Ⅎ 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( 𝜑 → ∀ 𝑥 𝜓 ) ) )