Theorem ralim

Description: Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997) (Proof shortened by Wolf Lammen, 1-Dec-2019)

	Ref	Expression
Assertion	ralim	$$⊢\forall x\in A\left(\phi \to \psi \right)\to \left(\forall x\in A\phi \to \forall x\in A\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		id	$$⊢\left(\phi \to \psi \right)\to \left(\phi \to \psi \right)$$
2	1	ral2imi	$$⊢\forall x\in A\left(\phi \to \psi \right)\to \left(\forall x\in A\phi \to \forall x\in A\psi \right)$$