Theorem mooran1

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 ) )