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Mirrors > Home > ILE Home > Th. List > euor2 | GIF version |
Description: Introduce or eliminate a disjunct in a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
euor2 | ⊢ (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑 ∨ 𝜓) ↔ ∃!𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbe1 1495 | . . 3 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) | |
2 | 1 | hbn 1654 | . 2 ⊢ (¬ ∃𝑥𝜑 → ∀𝑥 ¬ ∃𝑥𝜑) |
3 | 19.8a 1590 | . . . 4 ⊢ (𝜑 → ∃𝑥𝜑) | |
4 | 3 | con3i 632 | . . 3 ⊢ (¬ ∃𝑥𝜑 → ¬ 𝜑) |
5 | orel1 725 | . . . 4 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) → 𝜓)) | |
6 | olc 711 | . . . 4 ⊢ (𝜓 → (𝜑 ∨ 𝜓)) | |
7 | 5, 6 | impbid1 142 | . . 3 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓)) |
8 | 4, 7 | syl 14 | . 2 ⊢ (¬ ∃𝑥𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓)) |
9 | 2, 8 | eubidh 2032 | 1 ⊢ (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑 ∨ 𝜓) ↔ ∃!𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 105 ∨ wo 708 ∃wex 1492 ∃!weu 2026 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-4 1510 ax-17 1526 ax-ial 1534 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-fal 1359 df-eu 2029 |
This theorem is referenced by: reuun2 3418 |
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