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	$$⊢Ⅎx\phi$$
	ralrimia.2	$$⊢\left(\phi \wedge x\in A\right)\to \psi$$
Assertion	ralrimia	$$⊢\phi \to \forall x\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		ralrimia.1	$$⊢Ⅎx\phi$$
2		ralrimia.2	$$⊢\left(\phi \wedge x\in A\right)\to \psi$$
3	2	ex	$$⊢\phi \to \left(x\in A\to \psi \right)$$
4	1 3	ralrimi	$$⊢\phi \to \forall x\in A\psi$$