Theorem hbnd

Description: Deduction form of bound-variable hypothesis builder hbn . (Contributed by NM, 3-Jan-2002)

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

Proof

Step	Hyp	Ref	Expression
1		hbnd.1	$$⊢\phi \to \forall x\phi$$
2		hbnd.2	$$⊢\phi \to \left(\psi \to \forall x\psi \right)$$
3	1 2	alrimih	$$⊢\phi \to \forall x\left(\psi \to \forall x\psi \right)$$
4		hbnt	$$⊢\forall x\left(\psi \to \forall x\psi \right)\to \left(\neg \psi \to \forall x\neg \psi \right)$$
5	3 4	syl	$$⊢\phi \to \left(\neg \psi \to \forall x\neg \psi \right)$$