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Theorem ralrp 12943
Description: Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
Assertion
Ref Expression
ralrp (∀𝑥 ∈ ℝ+ 𝜑 ↔ ∀𝑥 ∈ ℝ (0 < 𝑥𝜑))

Proof of Theorem ralrp
StepHypRef Expression
1 elrp 12925 . . . 4 (𝑥 ∈ ℝ+ ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
21imbi1i 350 . . 3 ((𝑥 ∈ ℝ+𝜑) ↔ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) → 𝜑))
3 impexp 452 . . 3 (((𝑥 ∈ ℝ ∧ 0 < 𝑥) → 𝜑) ↔ (𝑥 ∈ ℝ → (0 < 𝑥𝜑)))
42, 3bitri 275 . 2 ((𝑥 ∈ ℝ+𝜑) ↔ (𝑥 ∈ ℝ → (0 < 𝑥𝜑)))
54ralbii2 3089 1 (∀𝑥 ∈ ℝ+ 𝜑 ↔ ∀𝑥 ∈ ℝ (0 < 𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107  wral 3061   class class class wbr 5109  cr 11058  0cc0 11059   < clt 11197  +crp 12923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-rp 12924
This theorem is referenced by: (None)
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