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Mirrors > Home > ILE Home > Th. List > euor | GIF version |
Description: Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) |
Ref | Expression |
---|---|
euor.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
Ref | Expression |
---|---|
euor | ⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euor.1 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | 1 | hbn 1544 | . . 3 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑) |
3 | biorf 663 | . . 3 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓))) | |
4 | 2, 3 | eubidh 1906 | . 2 ⊢ (¬ 𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑 ∨ 𝜓))) |
5 | 4 | biimpa 280 | 1 ⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ∨ wo 629 ∀wal 1241 ∃!weu 1900 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-17 1419 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-fal 1249 df-eu 1903 |
This theorem is referenced by: euorv 1927 |
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