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| Mirrors > Home > MPE Home > Th. List > euor2 | Structured version Visualization version GIF version | ||
| Description: Introduce or eliminate a disjunct in a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.) |
| Ref | Expression |
|---|---|
| euor2 | ⊢ (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑 ∨ 𝜓) ↔ ∃!𝑥𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfe1 2150 | . . 3 ⊢ Ⅎ𝑥∃𝑥𝜑 | |
| 2 | 1 | nfn 1857 | . 2 ⊢ Ⅎ𝑥 ¬ ∃𝑥𝜑 |
| 3 | 19.8a 2181 | . . 3 ⊢ (𝜑 → ∃𝑥𝜑) | |
| 4 | biorf 937 | . . . 4 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓))) | |
| 5 | 4 | bicomd 223 | . . 3 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓)) |
| 6 | 3, 5 | nsyl5 159 | . 2 ⊢ (¬ ∃𝑥𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓)) |
| 7 | 2, 6 | eubid 2587 | 1 ⊢ (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑 ∨ 𝜓) ↔ ∃!𝑥𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∨ wo 848 ∃wex 1779 ∃!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-10 2141 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-mo 2540 df-eu 2569 |
| This theorem is referenced by: reuun2 4325 |
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