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Metamath Proof Explorer
Description: "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004) (Proof shortened by Andrew Salmon, 9-Jul-2011)
|
|
Ref |
Expression |
|
Assertion |
mooran1 |
⊢ ( ( ∃* 𝑥 𝜑 ∨ ∃* 𝑥 𝜓 ) → ∃* 𝑥 ( 𝜑 ∧ 𝜓 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜑 ) |
| 2 |
1
|
moimi |
⊢ ( ∃* 𝑥 𝜑 → ∃* 𝑥 ( 𝜑 ∧ 𝜓 ) ) |
| 3 |
|
moan |
⊢ ( ∃* 𝑥 𝜓 → ∃* 𝑥 ( 𝜑 ∧ 𝜓 ) ) |
| 4 |
2 3
|
jaoi |
⊢ ( ( ∃* 𝑥 𝜑 ∨ ∃* 𝑥 𝜓 ) → ∃* 𝑥 ( 𝜑 ∧ 𝜓 ) ) |