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	|- ( ph -> A. x ph )
	hbim.2	|- ( ps -> A. x ps )
Assertion	hbim	|- ( ( ph -> ps ) -> A. x ( ph -> ps ) )

Proof

Step	Hyp	Ref	Expression
1		hbim.1	|- ( ph -> A. x ph )
2		hbim.2	|- ( ps -> A. x ps )
3	2	a1i	|- ( ph -> ( ps -> A. x ps ) )
4	1 3	hbim1	|- ( ( ph -> ps ) -> A. x ( ph -> ps ) )