Theorem ralrp

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

	Ref	Expression
Assertion	ralrp	$$⊢\forall x\in {ℝ}^{+}\phi \leftrightarrow \forall x\in ℝ\left(0<x\to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		elrp	$$⊢x\in {ℝ}^{+}\leftrightarrow \left(x\in ℝ\wedge 0<x\right)$$
2	1	imbi1i	$$⊢\left(x\in {ℝ}^{+}\to \phi \right)\leftrightarrow \left(\left(x\in ℝ\wedge 0<x\right)\to \phi \right)$$
3		impexp	$$⊢\left(\left(x\in ℝ\wedge 0<x\right)\to \phi \right)\leftrightarrow \left(x\in ℝ\to \left(0<x\to \phi \right)\right)$$
4	2 3	bitri	$$⊢\left(x\in {ℝ}^{+}\to \phi \right)\leftrightarrow \left(x\in ℝ\to \left(0<x\to \phi \right)\right)$$
5	4	ralbii2	$$⊢\forall x\in {ℝ}^{+}\phi \leftrightarrow \forall x\in ℝ\left(0<x\to \phi \right)$$