Description: Quantification over positive reals. (Contributed by NM, 12-Feb-2008)
Ref | Expression | ||
---|---|---|---|
Assertion | ralrp | ⊢ ( ∀ 𝑥 ∈ ℝ+ 𝜑 ↔ ∀ 𝑥 ∈ ℝ ( 0 < 𝑥 → 𝜑 ) ) |
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 ) ) |