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Theorem mopick2 1980
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 1519. (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 hbmo1 1935 . . . 4 (∃*xφx∃*xφ)
2 hbe1 1381 . . . 4 (x(φ ψ) → xx(φ ψ))
31, 2hban 1436 . . 3 ((∃*xφ x(φ ψ)) → x(∃*xφ x(φ ψ)))
4 mopick 1975 . . . . . 6 ((∃*xφ x(φ ψ)) → (φψ))
54ancld 308 . . . . 5 ((∃*xφ x(φ ψ)) → (φ → (φ ψ)))
65anim1d 319 . . . 4 ((∃*xφ x(φ ψ)) → ((φ χ) → ((φ ψ) χ)))
7 df-3an 886 . . . 4 ((φ ψ χ) ↔ ((φ ψ) χ))
86, 7syl6ibr 151 . . 3 ((∃*xφ x(φ ψ)) → ((φ χ) → (φ ψ χ)))
93, 8eximdh 1499 . 2 ((∃*xφ x(φ ψ)) → (x(φ χ) → x(φ ψ χ)))
1093impia 1100 1 ((∃*xφ x(φ ψ) x(φ χ)) → x(φ ψ χ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884  wex 1378  ∃*wmo 1898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-3an 886  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901
This theorem is referenced by: (None)
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