Theorem rexrp 13027

Description: Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)

Assertion

Ref	Expression
rexrp	⊢ (∃𝑥 ∈ ℝ+ 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ 𝜑))

Proof of Theorem rexrp

Step	Hyp	Ref	Expression
1		elrp 13008	. . . 4 ⊢ (𝑥 ∈ ℝ+ ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
2	1	anbi1i 623	. . 3 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝜑) ↔ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ 𝜑))
3		anass 468	. . 3 ⊢ (((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ 𝜑) ↔ (𝑥 ∈ ℝ ∧ (0 < 𝑥 ∧ 𝜑)))
4	2, 3	bitri 275	. 2 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝜑) ↔ (𝑥 ∈ ℝ ∧ (0 < 𝑥 ∧ 𝜑)))
5	4	rexbii2 3087	1 ⊢ (∃𝑥 ∈ ℝ+ 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ 𝜑))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∈ wcel 2099  ∃wrex 3067   class class class wbr 5148  ℝcr 11137  0cc0 11138   < clt 11278  ℝ⁺crp 13006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-rp 13007

This theorem is referenced by: (None)