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Theorem rexrp 8889
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 8869 . . . 4 (𝑥 ∈ ℝ+ ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
21anbi1i 446 . . 3 ((𝑥 ∈ ℝ+𝜑) ↔ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ 𝜑))
3 anass 393 . . 3 (((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ 𝜑) ↔ (𝑥 ∈ ℝ ∧ (0 < 𝑥𝜑)))
42, 3bitri 182 . 2 ((𝑥 ∈ ℝ+𝜑) ↔ (𝑥 ∈ ℝ ∧ (0 < 𝑥𝜑)))
54rexbii2 2382 1 (∃𝑥 ∈ ℝ+ 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wcel 1434  wrex 2354   class class class wbr 3805  cr 7094  0cc0 7095   < clt 7267  +crp 8867
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-rab 2362  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-rp 8868
This theorem is referenced by: (None)
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