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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 1664 | . . 3 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑) |
3 | biorf 745 | . . 3 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓))) | |
4 | 2, 3 | eubidh 2042 | . 2 ⊢ (¬ 𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑 ∨ 𝜓))) |
5 | 4 | biimpa 296 | 1 ⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 709 ∀wal 1361 ∃!weu 2036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-4 1520 ax-17 1536 ax-ial 1544 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-fal 1369 df-eu 2039 |
This theorem is referenced by: euorv 2063 |
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