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Theorem rexrp 9708
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 9687 . . . 4 (𝑥 ∈ ℝ+ ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
21anbi1i 458 . . 3 ((𝑥 ∈ ℝ+𝜑) ↔ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ 𝜑))
3 anass 401 . . 3 (((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ 𝜑) ↔ (𝑥 ∈ ℝ ∧ (0 < 𝑥𝜑)))
42, 3bitri 184 . 2 ((𝑥 ∈ ℝ+𝜑) ↔ (𝑥 ∈ ℝ ∧ (0 < 𝑥𝜑)))
54rexbii2 2501 1 (∃𝑥 ∈ ℝ+ 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wcel 2160  wrex 2469   class class class wbr 4018  cr 7841  0cc0 7842   < clt 8023  +crp 9685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-rab 2477  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-rp 9686
This theorem is referenced by: (None)
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