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Mirrors > Home > ILE Home > Th. List > rexrp | 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 9582 | . . . 4 ⊢ (𝑥 ∈ ℝ+ ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) | |
2 | 1 | anbi1i 454 | . . 3 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝜑) ↔ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ 𝜑)) |
3 | anass 399 | . . 3 ⊢ (((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ 𝜑) ↔ (𝑥 ∈ ℝ ∧ (0 < 𝑥 ∧ 𝜑))) | |
4 | 2, 3 | bitri 183 | . 2 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝜑) ↔ (𝑥 ∈ ℝ ∧ (0 < 𝑥 ∧ 𝜑))) |
5 | 4 | rexbii2 2475 | 1 ⊢ (∃𝑥 ∈ ℝ+ 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∈ wcel 2135 ∃wrex 2443 class class class wbr 3976 ℝcr 7743 0cc0 7744 < clt 7924 ℝ+crp 9580 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-rab 2451 df-v 2723 df-un 3115 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-rp 9581 |
This theorem is referenced by: (None) |
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