Theorem ralrp

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

	Ref	Expression
Assertion	ralrp	⊢ ( ∀ 𝑥 ∈ ℝ+ 𝜑 ↔ ∀ 𝑥 ∈ ℝ ( 0 < 𝑥 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		elrp	⊢ ( 𝑥 ∈ ℝ+ ↔ ( 𝑥 ∈ ℝ ∧ 0 < 𝑥 ) )
2	1	imbi1i	⊢ ( ( 𝑥 ∈ ℝ+ → 𝜑 ) ↔ ( ( 𝑥 ∈ ℝ ∧ 0 < 𝑥 ) → 𝜑 ) )
3		impexp	⊢ ( ( ( 𝑥 ∈ ℝ ∧ 0 < 𝑥 ) → 𝜑 ) ↔ ( 𝑥 ∈ ℝ → ( 0 < 𝑥 → 𝜑 ) ) )
4	2 3	bitri	⊢ ( ( 𝑥 ∈ ℝ+ → 𝜑 ) ↔ ( 𝑥 ∈ ℝ → ( 0 < 𝑥 → 𝜑 ) ) )
5	4	ralbii2	⊢ ( ∀ 𝑥 ∈ ℝ+ 𝜑 ↔ ∀ 𝑥 ∈ ℝ ( 0 < 𝑥 → 𝜑 ) )