Theorem rexrp 12056

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 12037	. . . 4 ⊢ (𝑥 ∈ ℝ+ ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
2	1	anbi1i 610	. . 3 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝜑) ↔ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ 𝜑))
3		anass 459	. . 3 ⊢ (((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ 𝜑) ↔ (𝑥 ∈ ℝ ∧ (0 < 𝑥 ∧ 𝜑)))
4	2, 3	bitri 264	. 2 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝜑) ↔ (𝑥 ∈ ℝ ∧ (0 < 𝑥 ∧ 𝜑)))
5	4	rexbii2 3187	1 ⊢ (∃𝑥 ∈ ℝ+ 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ 𝜑))

Syntax hints:   ↔ wb 196   ∧ wa 382   ∈ wcel 2145  ∃wrex 3062   class class class wbr 4786  ℝcr 10137  0cc0 10138   < clt 10276  ℝ⁺crp 12035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-rp 12036

This theorem is referenced by: (None)