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Theorem ralrp 12750
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 12732 . . . 4 (𝑥 ∈ ℝ+ ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
21imbi1i 350 . . 3 ((𝑥 ∈ ℝ+𝜑) ↔ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) → 𝜑))
3 impexp 451 . . 3 (((𝑥 ∈ ℝ ∧ 0 < 𝑥) → 𝜑) ↔ (𝑥 ∈ ℝ → (0 < 𝑥𝜑)))
42, 3bitri 274 . 2 ((𝑥 ∈ ℝ+𝜑) ↔ (𝑥 ∈ ℝ → (0 < 𝑥𝜑)))
54ralbii2 3090 1 (∀𝑥 ∈ ℝ+ 𝜑 ↔ ∀𝑥 ∈ ℝ (0 < 𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  wral 3064   class class class wbr 5074  cr 10870  0cc0 10871   < clt 11009  +crp 12730
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-rp 12731
This theorem is referenced by: (None)
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