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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 2084 | . 2 ⊢ (∃*𝑥(𝜑 ∨ 𝜓) → ∃*𝑥𝜑) | |
2 | olc 701 | . . 3 ⊢ (𝜓 → (𝜑 ∨ 𝜓)) | |
3 | 2 | moimi 2078 | . 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 698 ∃*wmo 2014 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 |
This theorem is referenced by: (None) |
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