Theorem rexrp 13038

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 13017	. . . 4 ⊢ (𝑥 ∈ ℝ+ ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
2	1	anbi1i 635	. . 3 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝜑) ↔ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ 𝜑))
3		anass 473	. . 3 ⊢ (((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ 𝜑) ↔ (𝑥 ∈ ℝ ∧ (0 < 𝑥 ∧ 𝜑)))
4	2, 3	bitri 278	. 2 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝜑) ↔ (𝑥 ∈ ℝ ∧ (0 < 𝑥 ∧ 𝜑)))
5	4	rexbii2 3114	1 ⊢ (∃𝑥 ∈ ℝ+ 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ 𝜑))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∈ wcel 2149  ∃wrex 3095   class class class wbr 5113  ℝcr 11098  0cc0 11099   < clt 11242  ℝ⁺crp 13015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-rp 13016

This theorem is referenced by: (None)