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Theorem rexrp 12996
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 12977 . . . 4 (𝑥 ∈ ℝ+ ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
21anbi1i 623 . . 3 ((𝑥 ∈ ℝ+𝜑) ↔ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ 𝜑))
3 anass 468 . . 3 (((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ 𝜑) ↔ (𝑥 ∈ ℝ ∧ (0 < 𝑥𝜑)))
42, 3bitri 275 . 2 ((𝑥 ∈ ℝ+𝜑) ↔ (𝑥 ∈ ℝ ∧ (0 < 𝑥𝜑)))
54rexbii2 3082 1 (∃𝑥 ∈ ℝ+ 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wcel 2098  wrex 3062   class class class wbr 5139  cr 11106  0cc0 11107   < clt 11247  +crp 12975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-rp 12976
This theorem is referenced by: (None)
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