Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ralrp | 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 9436 | . . . 4 ⊢ (𝑥 ∈ ℝ+ ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) | |
2 | 1 | imbi1i 237 | . . 3 ⊢ ((𝑥 ∈ ℝ+ → 𝜑) ↔ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) → 𝜑)) |
3 | impexp 261 | . . 3 ⊢ (((𝑥 ∈ ℝ ∧ 0 < 𝑥) → 𝜑) ↔ (𝑥 ∈ ℝ → (0 < 𝑥 → 𝜑))) | |
4 | 2, 3 | bitri 183 | . 2 ⊢ ((𝑥 ∈ ℝ+ → 𝜑) ↔ (𝑥 ∈ ℝ → (0 < 𝑥 → 𝜑))) |
5 | 4 | ralbii2 2443 | 1 ⊢ (∀𝑥 ∈ ℝ+ 𝜑 ↔ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1480 ∀wral 2414 class class class wbr 3924 ℝcr 7612 0cc0 7613 < clt 7793 ℝ+crp 9434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rab 2423 df-v 2683 df-un 3070 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-rp 9435 |
This theorem is referenced by: caucvgre 10746 |
Copyright terms: Public domain | W3C validator |