This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )