Theorem rexrp

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

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

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 < 𝑥 ∧ 𝜑 ) ) )
4	2 3	bitri	⊢ ( ( 𝑥 ∈ ℝ+ ∧ 𝜑 ) ↔ ( 𝑥 ∈ ℝ ∧ ( 0 < 𝑥 ∧ 𝜑 ) ) )
5	4	rexbii2	⊢ ( ∃ 𝑥 ∈ ℝ+ 𝜑 ↔ ∃ 𝑥 ∈ ℝ ( 0 < 𝑥 ∧ 𝜑 ) )