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Metamath Proof Explorer

Theorem alimd

Description: Deduction form of Theorem 19.20 of Margaris p. 90, see alim . See alimdh , alimdv for variants requiring fewer axioms. (Contributed by Mario Carneiro, 24-Sep-2016)

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

Proof

Step	Hyp	Ref	Expression
1		alimd.1	⊢ Ⅎ 𝑥 𝜑
2		alimd.2	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
3	1	nf5ri	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
4	3 2	alimdh	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑥 𝜒 ) )