Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ralrp | Structured version Visualization version GIF version |
Description: Quantification over positive reals. (Contributed by NM, 12-Feb-2008.) |
Ref | Expression |
---|---|
ralrp | ⊢ (∀𝑥 ∈ ℝ+ 𝜑 ↔ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrp 12474 | . . . 4 ⊢ (𝑥 ∈ ℝ+ ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) | |
2 | 1 | imbi1i 353 | . . 3 ⊢ ((𝑥 ∈ ℝ+ → 𝜑) ↔ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) → 𝜑)) |
3 | impexp 454 | . . 3 ⊢ (((𝑥 ∈ ℝ ∧ 0 < 𝑥) → 𝜑) ↔ (𝑥 ∈ ℝ → (0 < 𝑥 → 𝜑))) | |
4 | 2, 3 | bitri 278 | . 2 ⊢ ((𝑥 ∈ ℝ+ → 𝜑) ↔ (𝑥 ∈ ℝ → (0 < 𝑥 → 𝜑))) |
5 | 4 | ralbii2 3078 | 1 ⊢ (∀𝑥 ∈ ℝ+ 𝜑 ↔ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2114 ∀wral 3053 class class class wbr 5030 ℝcr 10614 0cc0 10615 < clt 10753 ℝ+crp 12472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rab 3062 df-v 3400 df-un 3848 df-sn 4517 df-pr 4519 df-op 4523 df-br 5031 df-rp 12473 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |