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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 2151 | . . 3 ⊢ Ⅎ𝑥∃𝑥𝜑 | |
2 | 1 | nfn 1858 | . 2 ⊢ Ⅎ𝑥 ¬ ∃𝑥𝜑 |
3 | 19.8a 2178 | . . 3 ⊢ (𝜑 → ∃𝑥𝜑) | |
4 | biorf 934 | . . . 4 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓))) | |
5 | 4 | bicomd 226 | . . 3 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓)) |
6 | 3, 5 | nsyl5 162 | . 2 ⊢ (¬ ∃𝑥𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓)) |
7 | 2, 6 | eubid 2648 | 1 ⊢ (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑 ∨ 𝜓) ↔ ∃!𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∨ wo 844 ∃wex 1781 ∃!weu 2628 |
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-10 2142 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-nf 1786 df-mo 2598 df-eu 2629 |
This theorem is referenced by: reuun2 4238 |
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