Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  rexrp GIF version

Theorem rexrp 9489
 Description: Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)
Assertion
Ref Expression
rexrp (∃𝑥 ∈ ℝ+ 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥𝜑))

Proof of Theorem rexrp
StepHypRef Expression
1 elrp 9468 . . . 4 (𝑥 ∈ ℝ+ ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
21anbi1i 454 . . 3 ((𝑥 ∈ ℝ+𝜑) ↔ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ 𝜑))
3 anass 399 . . 3 (((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ 𝜑) ↔ (𝑥 ∈ ℝ ∧ (0 < 𝑥𝜑)))
42, 3bitri 183 . 2 ((𝑥 ∈ ℝ+𝜑) ↔ (𝑥 ∈ ℝ ∧ (0 < 𝑥𝜑)))
54rexbii2 2447 1 (∃𝑥 ∈ ℝ+ 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥𝜑))
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104   ∈ wcel 1481  ∃wrex 2418   class class class wbr 3933  ℝcr 7639  0cc0 7640   < clt 7820  ℝ+crp 9466 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-rab 2426  df-v 2689  df-un 3076  df-sn 3534  df-pr 3535  df-op 3537  df-br 3934  df-rp 9467 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator