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 )