Metamath Proof Explorer


Theorem rexrp

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

Ref Expression
Assertion rexrp ( ∃ 𝑥 ∈ ℝ+ 𝜑 ↔ ∃ 𝑥 ∈ ℝ ( 0 < 𝑥𝜑 ) )

Proof

Step Hyp Ref Expression
1 elrp ( 𝑥 ∈ ℝ+ ↔ ( 𝑥 ∈ ℝ ∧ 0 < 𝑥 ) )
2 1 anbi1i ( ( 𝑥 ∈ ℝ+𝜑 ) ↔ ( ( 𝑥 ∈ ℝ ∧ 0 < 𝑥 ) ∧ 𝜑 ) )
3 anass ( ( ( 𝑥 ∈ ℝ ∧ 0 < 𝑥 ) ∧ 𝜑 ) ↔ ( 𝑥 ∈ ℝ ∧ ( 0 < 𝑥𝜑 ) ) )
4 2 3 bitri ( ( 𝑥 ∈ ℝ+𝜑 ) ↔ ( 𝑥 ∈ ℝ ∧ ( 0 < 𝑥𝜑 ) ) )
5 4 rexbii2 ( ∃ 𝑥 ∈ ℝ+ 𝜑 ↔ ∃ 𝑥 ∈ ℝ ( 0 < 𝑥𝜑 ) )