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Metamath Proof Explorer

Theorem ralrimia

Description: Inference from Theorem 19.21 of Margaris p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses ralrimia.1 ⊢ Ⅎ x φ
ralrimia.2 ⊢ φ ∧ x ∈ A → ψ
Assertion ralrimia ⊢ φ → ∀ x ∈ A ψ

Proof

Step Hyp Ref Expression
1 ralrimia.1 ⊢ Ⅎ x φ
2 ralrimia.2 ⊢ φ ∧ x ∈ A → ψ
3 2 ex ⊢ φ → x ∈ A → ψ
4 1 3 ralrimi ⊢ φ → ∀ x ∈ A ψ