Theorem ralrp

Description: Quantification over positive reals. (Contributed by NM, 12-Feb-2008)

	Ref	Expression
Assertion	ralrp	|- ( A. x e. RR+ ph <-> A. x e. RR ( 0 < x -> ph ) )

Proof

Step	Hyp	Ref	Expression
1		elrp	|- ( x e. RR+ <-> ( x e. RR /\ 0 < x ) )
2	1	imbi1i	|- ( ( x e. RR+ -> ph ) <-> ( ( x e. RR /\ 0 < x ) -> ph ) )
3		impexp	|- ( ( ( x e. RR /\ 0 < x ) -> ph ) <-> ( x e. RR -> ( 0 < x -> ph ) ) )
4	2 3	bitri	|- ( ( x e. RR+ -> ph ) <-> ( x e. RR -> ( 0 < x -> ph ) ) )
5	4	ralbii2	|- ( A. x e. RR+ ph <-> A. x e. RR ( 0 < x -> ph ) )