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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 2655 | . . . 4 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) | |
2 | moanim.1 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
3 | simpl 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
4 | 2, 3 | exlimi 2207 | . . . 4 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → 𝜑) |
6 | ibar 529 | . . . . 5 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓))) | |
7 | 2, 6 | eubid 2666 | . . . 4 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑 ∧ 𝜓))) |
8 | 7 | biimprcd 251 | . . 3 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → (𝜑 → ∃!𝑥𝜓)) |
9 | 5, 8 | jcai 517 | . 2 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → (𝜑 ∧ ∃!𝑥𝜓)) |
10 | 7 | biimpa 477 | . 2 ⊢ ((𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∧ 𝜓)) |
11 | 9, 10 | impbii 210 | 1 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∃wex 1771 Ⅎwnf 1775 ∃!weu 2646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 df-mo 2615 df-eu 2647 |
This theorem is referenced by: 2eu7 2738 2eu8 2739 |
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