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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
StepHypRef Expression
1 nfmo1 2215 . . . 4 x∃*xφ
2 nfe1 1732 . . . 4 xx(φ ψ)
31, 2nfan 1824 . . 3 x(∃*xφ x(φ ψ))
4 mopick 2266 . . . . . 6 ((∃*xφ x(φ ψ)) → (φψ))
54ancld 536 . . . . 5 ((∃*xφ x(φ ψ)) → (φ → (φ ψ)))
65anim1d 547 . . . 4 ((∃*xφ x(φ ψ)) → ((φ χ) → ((φ ψ) χ)))
7 df-3an 936 . . . 4 ((φ ψ χ) ↔ ((φ ψ) χ))
86, 7syl6ibr 218 . . 3 ((∃*xφ x(φ ψ)) → ((φ χ) → (φ ψ χ)))
93, 8eximd 1770 . 2 ((∃*xφ x(φ ψ)) → (x(φ χ) → x(φ ψ χ)))
1093impia 1148 1 ((∃*xφ x(φ ψ) x(φ χ)) → x(φ ψ χ))
 Colors of variables: wff setvar class 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)
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