Theorem alimdh

Description: Deduction form of Theorem 19.20 of Margaris p. 90, see alim . (Contributed by NM, 4-Jan-2002)

	Ref	Expression
Hypotheses	alimdh.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	alimdh.2	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
Assertion	alimdh	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑥 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		alimdh.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		alimdh.2	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
3	2	al2imi	⊢ ( ∀ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑥 𝜒 ) )
4	1 3	syl	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑥 𝜒 ) )