Theorem ralrimdvva

Description: Inference from Theorem 19.21 of Margaris p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 2-Feb-2008)

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

Proof

Step	Hyp	Ref	Expression
1		ralrimdvva.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝜓 → 𝜒 ) )
2	1	ex	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ( 𝜓 → 𝜒 ) ) )
3	2	com23	⊢ ( 𝜑 → ( 𝜓 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → 𝜒 ) ) )
4	3	ralrimdvv	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜒 ) )