Theorem hbim

Description: If x is not free in ph and ps , it is not free in ( ph -> ps ) . (Contributed by NM, 24-Jan-1993) (Proof shortened by Mel L. O'Cat, 3-Mar-2008) (Proof shortened by Wolf Lammen, 1-Jan-2018)

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

Proof

Step	Hyp	Ref	Expression
1		hbim.1	$$⊢\phi \to \forall x\phi$$
2		hbim.2	$$⊢\psi \to \forall x\psi$$
3	2	a1i	$$⊢\phi \to \left(\psi \to \forall x\psi \right)$$
4	1 3	hbim1	$$⊢\left(\phi \to \psi \right)\to \forall x\left(\phi \to \psi \right)$$