Metamath Proof Explorer


Theorem mooran2

Description: "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004) (Proof shortened by Andrew Salmon, 9-Jul-2011)

Ref Expression
Assertion mooran2 ( ∃* 𝑥 ( 𝜑𝜓 ) → ( ∃* 𝑥 𝜑 ∧ ∃* 𝑥 𝜓 ) )

Proof

Step Hyp Ref Expression
1 moor ( ∃* 𝑥 ( 𝜑𝜓 ) → ∃* 𝑥 𝜑 )
2 olc ( 𝜓 → ( 𝜑𝜓 ) )
3 2 moimi ( ∃* 𝑥 ( 𝜑𝜓 ) → ∃* 𝑥 𝜓 )
4 1 3 jca ( ∃* 𝑥 ( 𝜑𝜓 ) → ( ∃* 𝑥 𝜑 ∧ ∃* 𝑥 𝜓 ) )