Theorem ralrimdva

Description: Inference from Theorem 19.21 of Margaris p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Feb-2008) (Proof shortened by Wolf Lammen, 28-Dec-2019)

	Ref	Expression
Hypothesis	ralrimdva.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 → 𝜒 ) )
Assertion	ralrimdva	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 ∈ 𝐴 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ralrimdva.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 → 𝜒 ) )
2	1	expimpd	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) → 𝜒 ) )
3	2	expcomd	⊢ ( 𝜑 → ( 𝜓 → ( 𝑥 ∈ 𝐴 → 𝜒 ) ) )
4	3	ralrimdv	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 ∈ 𝐴 𝜒 ) )