Theorem ralrp 12962

Description: Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)

Assertion

Ref	Expression
ralrp	⊢ (∀𝑥 ∈ ℝ+ 𝜑 ↔ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝜑))

Proof of Theorem ralrp

Step	Hyp	Ref	Expression
1		elrp 12942	. . . 4 ⊢ (𝑥 ∈ ℝ+ ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
2	1	imbi1i 350	. . 3 ⊢ ((𝑥 ∈ ℝ+ → 𝜑) ↔ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) → 𝜑))
3		impexp 451	. . 3 ⊢ (((𝑥 ∈ ℝ ∧ 0 < 𝑥) → 𝜑) ↔ (𝑥 ∈ ℝ → (0 < 𝑥 → 𝜑)))
4	2, 3	bitri 276	. 2 ⊢ ((𝑥 ∈ ℝ+ → 𝜑) ↔ (𝑥 ∈ ℝ → (0 < 𝑥 → 𝜑)))
5	4	ralbii2 3082	1 ⊢ (∀𝑥 ∈ ℝ+ 𝜑 ↔ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∈ wcel 2119  ∀wral 3054   class class class wbr 5079  ℝcr 11035  0cc0 11036   < clt 11177  ℝ⁺crp 12940

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-rp 12941

This theorem is referenced by: (None)