Theorem mopick2

Description: "At most one" can show the existence of a common value. In this case we can infer existence of conjunction from a conjunction of existence, and it is one way to achieve the converse of 19.40 . (Contributed by NM, 5-Apr-2004) (Proof shortened by Andrew Salmon, 9-Jul-2011)

	Ref	Expression
Assertion	mopick2	⊢ ( ( ∃* 𝑥 𝜑 ∧ ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑥 ( 𝜑 ∧ 𝜒 ) ) → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		nfmo1	⊢ Ⅎ 𝑥 ∃* 𝑥 𝜑
2		nfe1	⊢ Ⅎ 𝑥 ∃ 𝑥 ( 𝜑 ∧ 𝜓 )
3	1 2	nfan	⊢ Ⅎ 𝑥 ( ∃* 𝑥 𝜑 ∧ ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) )
4		mopick	⊢ ( ( ∃* 𝑥 𝜑 ∧ ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) → ( 𝜑 → 𝜓 ) )
5	4	ancld	⊢ ( ( ∃* 𝑥 𝜑 ∧ ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) → ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) )
6	5	anim1d	⊢ ( ( ∃* 𝑥 𝜑 ∧ ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) → ( ( 𝜑 ∧ 𝜒 ) → ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ) )
7		df-3an	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) )
8	6 7	syl6ibr	⊢ ( ( ∃* 𝑥 𝜑 ∧ ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) → ( ( 𝜑 ∧ 𝜒 ) → ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ) )
9	3 8	eximd	⊢ ( ( ∃* 𝑥 𝜑 ∧ ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) → ( ∃ 𝑥 ( 𝜑 ∧ 𝜒 ) → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ) )
10	9	3impia	⊢ ( ( ∃* 𝑥 𝜑 ∧ ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑥 ( 𝜑 ∧ 𝜒 ) ) → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) )