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Metamath Proof Explorer

Theorem ralim

Description: Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997) (Proof shortened by Wolf Lammen, 1-Dec-2019)

Ref Expression
Assertion ralim ⊢ ∀ x ∈ A φ → ψ → ∀ x ∈ A φ → ∀ x ∈ A ψ

Proof

Step Hyp Ref Expression
1 id ⊢ φ → ψ → φ → ψ
2 1 ral2imi ⊢ ∀ x ∈ A φ → ψ → ∀ x ∈ A φ → ∀ x ∈ A ψ