Theorem rexrp

Description: Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014)

	Ref	Expression
Assertion	rexrp	|- ( E. x e. RR+ ph <-> E. x e. RR ( 0 < x /\ ph ) )

Proof

Step	Hyp	Ref	Expression
1		elrp	|- ( x e. RR+ <-> ( x e. RR /\ 0 < x ) )
2	1	anbi1i	|- ( ( x e. RR+ /\ ph ) <-> ( ( x e. RR /\ 0 < x ) /\ ph ) )
3		anass	|- ( ( ( 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	rexbii2	|- ( E. x e. RR+ ph <-> E. x e. RR ( 0 < x /\ ph ) )