Theorem ralrimivv

Description: Inference from Theorem 19.21 of Margaris p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 24-Jul-2004)

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

Proof

Step	Hyp	Ref	Expression
1		ralrimivv.1	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → 𝜓 ) )
2	1	expd	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐵 → 𝜓 ) ) )
3	2	ralrimdv	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐵 𝜓 ) )
4	3	ralrimiv	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 )