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 ( 𝜑 ∨ 𝜓 ) → ( ∃* 𝑥 𝜑 ∧ ∃* 𝑥 𝜓 ) )

Proof

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