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Mirrors > Home > MPE Home > Th. List > euorv | Structured version Visualization version GIF version |
Description: Introduce a disjunct into a unique existential quantifier. Version of euor 2691 requiring disjoint variables, but fewer axioms. (Contributed by NM, 23-Mar-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2023.) |
Ref | Expression |
---|---|
euorv | ⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biorf 933 | . . 3 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓))) | |
2 | 1 | eubidv 2668 | . 2 ⊢ (¬ 𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑 ∨ 𝜓))) |
3 | 2 | biimpa 479 | 1 ⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 ∃!weu 2649 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1777 df-mo 2618 df-eu 2650 |
This theorem is referenced by: eueq2 3700 |
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