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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 uniqueness 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 2027 | . . 3 ⊢ Ⅎ𝑥∃𝑥𝜑 | |
2 | 1 | nfn 1784 | . 2 ⊢ Ⅎ𝑥 ¬ ∃𝑥𝜑 |
3 | 19.8a 2052 | . . . 4 ⊢ (𝜑 → ∃𝑥𝜑) | |
4 | 3 | con3i 150 | . . 3 ⊢ (¬ ∃𝑥𝜑 → ¬ 𝜑) |
5 | biorf 420 | . . . 4 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓))) | |
6 | 5 | bicomd 213 | . . 3 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓)) |
7 | 4, 6 | syl 17 | . 2 ⊢ (¬ ∃𝑥𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓)) |
8 | 2, 7 | eubid 2488 | 1 ⊢ (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑 ∨ 𝜓) ↔ ∃!𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∃wex 1704 ∃!weu 2470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-12 2047 |
This theorem depends on definitions: df-bi 197 df-or 385 df-ex 1705 df-nf 1710 df-eu 2474 |
This theorem is referenced by: reuun2 3910 |
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