Theorem mopick2 2271

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 1609. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

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

Proof of Theorem mopick2

Step	Hyp	Ref	Expression
1		nfmo1 2215	. . . 4 ⊢ Ⅎx∃*xφ
2		nfe1 1732	. . . 4 ⊢ Ⅎx∃x(φ ∧ ψ)
3	1, 2	nfan 1824	. . 3 ⊢ Ⅎx(∃*xφ ∧ ∃x(φ ∧ ψ))
4		mopick 2266	. . . . . 6 ⊢ ((∃*xφ ∧ ∃x(φ ∧ ψ)) → (φ → ψ))
5	4	ancld 536	. . . . 5 ⊢ ((∃*xφ ∧ ∃x(φ ∧ ψ)) → (φ → (φ ∧ ψ)))
6	5	anim1d 547	. . . 4 ⊢ ((∃*xφ ∧ ∃x(φ ∧ ψ)) → ((φ ∧ χ) → ((φ ∧ ψ) ∧ χ)))
7		df-3an 936	. . . 4 ⊢ ((φ ∧ ψ ∧ χ) ↔ ((φ ∧ ψ) ∧ χ))
8	6, 7	syl6ibr 218	. . 3 ⊢ ((∃*xφ ∧ ∃x(φ ∧ ψ)) → ((φ ∧ χ) → (φ ∧ ψ ∧ χ)))
9	3, 8	eximd 1770	. 2 ⊢ ((∃*xφ ∧ ∃x(φ ∧ ψ)) → (∃x(φ ∧ χ) → ∃x(φ ∧ ψ ∧ χ)))
10	9	3impia 1148	1 ⊢ ((∃*xφ ∧ ∃x(φ ∧ ψ) ∧ ∃x(φ ∧ χ)) → ∃x(φ ∧ ψ ∧ χ))

Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934  ∃wex 1541  ∃*wmo 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by: (None)