Theorem ralimdaa

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of Margaris p. 90. (Contributed by NM, 22-Sep-2003) (Proof shortened by Wolf Lammen, 29-Dec-2019)

	Ref	Expression
Hypotheses	ralimdaa.1	$$⊢Ⅎx\phi$$
	ralimdaa.2	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \to \chi \right)$$
Assertion	ralimdaa	$$⊢\phi \to \left(\forall x\in A\psi \to \forall x\in A\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ralimdaa.1	$$⊢Ⅎx\phi$$
2		ralimdaa.2	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \to \chi \right)$$
3	2	ex	$$⊢\phi \to \left(x\in A\to \left(\psi \to \chi \right)\right)$$
4	1 3	ralrimi	$$⊢\phi \to \forall x\in A\left(\psi \to \chi \right)$$
5		ralim	$$⊢\forall x\in A\left(\psi \to \chi \right)\to \left(\forall x\in A\psi \to \forall x\in A\chi \right)$$
6	4 5	syl	$$⊢\phi \to \left(\forall x\in A\psi \to \forall x\in A\chi \right)$$