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Mirrors > Home > ILE Home > Th. List > euor | GIF version |
Description: Introduce a disjunct into a unique existential 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 1632 | . . 3 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑) |
3 | biorf 733 | . . 3 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓))) | |
4 | 2, 3 | eubidh 2005 | . 2 ⊢ (¬ 𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑 ∨ 𝜓))) |
5 | 4 | biimpa 294 | 1 ⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 697 ∀wal 1329 ∃!weu 1999 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-17 1506 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-eu 2002 |
This theorem is referenced by: euorv 2026 |
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