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| Mirrors > Home > MPE Home > Th. List > mooran2 | Structured version Visualization version 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 2554 | . 2 ⊢ (∃*𝑥(𝜑 ∨ 𝜓) → ∃*𝑥𝜑) | |
| 2 | olc 869 | . . 3 ⊢ (𝜓 → (𝜑 ∨ 𝜓)) | |
| 3 | 2 | moimi 2545 | . 2 ⊢ (∃*𝑥(𝜑 ∨ 𝜓) → ∃*𝑥𝜓) |
| 4 | 1, 3 | jca 511 | 1 ⊢ (∃*𝑥(𝜑 ∨ 𝜓) → (∃*𝑥𝜑 ∧ ∃*𝑥𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 ∃*wmo 2538 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-mo 2540 |
| This theorem is referenced by: rmoun 32513 |
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