Theorem al2im

Description: Closed form of al2imi . Version of alim for a nested implication. (Contributed by Alan Sare, 31-Dec-2011)

	Ref	Expression
Assertion	al2im	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ∀ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑥 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		alim	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ∀ 𝑥 𝜑 → ∀ 𝑥 ( 𝜓 → 𝜒 ) ) )
2		alim	⊢ ( ∀ 𝑥 ( 𝜓 → 𝜒 ) → ( ∀ 𝑥 𝜓 → ∀ 𝑥 𝜒 ) )
3	1 2	syl6	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ∀ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑥 𝜒 ) ) )