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Theorem ralrp 12397
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 12379 . . . 4 (𝑥 ∈ ℝ+ ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
21imbi1i 353 . . 3 ((𝑥 ∈ ℝ+𝜑) ↔ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) → 𝜑))
3 impexp 454 . . 3 (((𝑥 ∈ ℝ ∧ 0 < 𝑥) → 𝜑) ↔ (𝑥 ∈ ℝ → (0 < 𝑥𝜑)))
42, 3bitri 278 . 2 ((𝑥 ∈ ℝ+𝜑) ↔ (𝑥 ∈ ℝ → (0 < 𝑥𝜑)))
54ralbii2 3155 1 (∀𝑥 ∈ ℝ+ 𝜑 ↔ ∀𝑥 ∈ ℝ (0 < 𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2114  wral 3130   class class class wbr 5042  cr 10525  0cc0 10526   < clt 10664  +crp 12377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rab 3139  df-v 3471  df-un 3913  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-rp 12378
This theorem is referenced by: (None)
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