Theorem ralrimdvv

Description: Inference from Theorem 19.21 of Margaris p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 1-Jun-2005)

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

Proof

Step	Hyp	Ref	Expression
1		ralrimdvv.1	⊢ ( 𝜑 → ( 𝜓 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → 𝜒 ) ) )
2	1	imp	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → 𝜒 ) )
3	2	ralrimivv	⊢ ( ( 𝜑 ∧ 𝜓 ) → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜒 )
4	3	ex	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜒 ) )