Theorem ralrimdv

Description: Inference from Theorem 19.21 of Margaris p. 90. (Restricted quantifier version.) (Contributed by NM, 27-May-1998) Reduce dependencies on axioms. (Revised by Wolf Lammen, 28-Dec-2019)

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

Proof

Step	Hyp	Ref	Expression
1		ralrimdv.1	⊢ ( 𝜑 → ( 𝜓 → ( 𝑥 ∈ 𝐴 → 𝜒 ) ) )
2	1	imp	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑥 ∈ 𝐴 → 𝜒 ) )
3	2	ralrimiv	⊢ ( ( 𝜑 ∧ 𝜓 ) → ∀ 𝑥 ∈ 𝐴 𝜒 )
4	3	ex	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 ∈ 𝐴 𝜒 ) )