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Mirrors > Home > MPE Home > Th. List > euor | Structured version Visualization version GIF version |
Description: Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) |
Ref | Expression |
---|---|
euor.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
euor | ⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euor.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | nfn 1784 | . . 3 ⊢ Ⅎ𝑥 ¬ 𝜑 |
3 | biorf 420 | . . 3 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓))) | |
4 | 2, 3 | eubid 2488 | . 2 ⊢ (¬ 𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑 ∨ 𝜓))) |
5 | 4 | biimpa 501 | 1 ⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 383 ∧ wa 384 Ⅎwnf 1708 ∃!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-12 2047 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-nf 1710 df-eu 2474 |
This theorem is referenced by: euorv 2513 |
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