Theorem hbim1

Description: A closed form of hbim . (Contributed by NM, 2-Jun-1993)

	Ref	Expression
Hypotheses	hbim1.1	$$⊢\phi \to \forall x\phi$$
	hbim1.2	$$⊢\phi \to \left(\psi \to \forall x\psi \right)$$
Assertion	hbim1	$$⊢\left(\phi \to \psi \right)\to \forall x\left(\phi \to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		hbim1.1	$$⊢\phi \to \forall x\phi$$
2		hbim1.2	$$⊢\phi \to \left(\psi \to \forall x\psi \right)$$
3	2	a2i	$$⊢\left(\phi \to \psi \right)\to \left(\phi \to \forall x\psi \right)$$
4	1	19.21h	$$⊢\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\psi \right)$$
5	3 4	sylibr	$$⊢\left(\phi \to \psi \right)\to \forall x\left(\phi \to \psi \right)$$