Metamath Proof Explorer


Theorem ralrp

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

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

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 < 𝑥𝜑 ) ) )
4 2 3 bitri ( ( 𝑥 ∈ ℝ+𝜑 ) ↔ ( 𝑥 ∈ ℝ → ( 0 < 𝑥𝜑 ) ) )
5 4 ralbii2 ( ∀ 𝑥 ∈ ℝ+ 𝜑 ↔ ∀ 𝑥 ∈ ℝ ( 0 < 𝑥𝜑 ) )