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Mirrors > Home > MPE Home > Th. List > rexrp | Structured version Visualization version GIF version |
Description: Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.) |
Ref | Expression |
---|---|
rexrp | ⊢ (∃𝑥 ∈ ℝ+ 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrp 12379 | . . . 4 ⊢ (𝑥 ∈ ℝ+ ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) | |
2 | 1 | anbi1i 626 | . . 3 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝜑) ↔ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ 𝜑)) |
3 | anass 472 | . . 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 3208 | 1 ⊢ (∃𝑥 ∈ ℝ+ 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2111 ∃wrex 3107 class class class wbr 5030 ℝcr 10525 0cc0 10526 < clt 10664 ℝ+crp 12377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rex 3112 df-rab 3115 df-v 3443 df-un 3886 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-rp 12378 |
This theorem is referenced by: (None) |
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