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Mirrors > Home > ILE Home > Th. List > mooran2 | GIF version |
Description: "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
mooran2 | ⊢ (∃*𝑥(𝜑 ∨ 𝜓) → (∃*𝑥𝜑 ∧ ∃*𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moor 2048 | . 2 ⊢ (∃*𝑥(𝜑 ∨ 𝜓) → ∃*𝑥𝜑) | |
2 | olc 685 | . . 3 ⊢ (𝜓 → (𝜑 ∨ 𝜓)) | |
3 | 2 | moimi 2042 | . 2 ⊢ (∃*𝑥(𝜑 ∨ 𝜓) → ∃*𝑥𝜓) |
4 | 1, 3 | jca 304 | 1 ⊢ (∃*𝑥(𝜑 ∨ 𝜓) → (∃*𝑥𝜑 ∧ ∃*𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ wo 682 ∃*wmo 1978 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 |
This theorem is referenced by: (None) |
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