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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 )