Theorem ralimdvv

Description: Deduction doubly quantifying both antecedent and consequent. (Contributed by Scott Fenton, 2-Mar-2025)

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

Proof

Step	Hyp	Ref	Expression
1		ralimdvv.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2	1	adantr	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝜓 → 𝜒 ) )
3	2	ralimdvva	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜒 ) )