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| Mirrors > Home > MPE Home > Th. List > euan | Structured version Visualization version GIF version | ||
| Description: Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 24-Dec-2018.) |
| Ref | Expression |
|---|---|
| moanim.1 | ⊢ Ⅎ𝑥𝜑 |
| Ref | Expression |
|---|---|
| euan | ⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | euex 2577 | . . . 4 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) | |
| 2 | moanim.1 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
| 3 | simpl 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
| 4 | 2, 3 | exlimi 2217 | . . . 4 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑) |
| 5 | 1, 4 | syl 17 | . . 3 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → 𝜑) |
| 6 | ibar 528 | . . . . 5 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓))) | |
| 7 | 2, 6 | eubid 2587 | . . . 4 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑 ∧ 𝜓))) |
| 8 | 7 | biimprcd 250 | . . 3 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → (𝜑 → ∃!𝑥𝜓)) |
| 9 | 5, 8 | jcai 516 | . 2 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → (𝜑 ∧ ∃!𝑥𝜓)) |
| 10 | 7 | biimpa 476 | . 2 ⊢ ((𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∧ 𝜓)) |
| 11 | 9, 10 | impbii 209 | 1 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∃wex 1779 Ⅎwnf 1783 ∃!weu 2568 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-mo 2540 df-eu 2569 |
| This theorem is referenced by: 2eu7 2658 2eu8 2659 |
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