Theorem 2ralimi

Description: Inference quantifying both antecedent and consequent two times, with strong hypothesis. (Contributed by AV, 3-Dec-2021)

	Ref	Expression
Hypothesis	2ralimi.1	⊢ ( 𝜑 → 𝜓 )
Assertion	2ralimi	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		2ralimi.1	⊢ ( 𝜑 → 𝜓 )
2	1	ralimi	⊢ ( ∀ 𝑦 ∈ 𝐵 𝜑 → ∀ 𝑦 ∈ 𝐵 𝜓 )
3	2	ralimi	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 )