Description: Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014)
Ref | Expression | ||
---|---|---|---|
Assertion | rexrp | ⊢ ( ∃ 𝑥 ∈ ℝ+ 𝜑 ↔ ∃ 𝑥 ∈ ℝ ( 0 < 𝑥 ∧ 𝜑 ) ) |
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 ) ) |