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Theorem ralrp 12492
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 12474 . . . 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 3078 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 3053   class class class wbr 5030  cr 10614  0cc0 10615   < clt 10753  +crp 12472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rab 3062  df-v 3400  df-un 3848  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-rp 12473
This theorem is referenced by: (None)
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