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Metamath Proof Explorer

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 )