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Theorem mopick2 2082
 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 1610. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
mopick2 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓) ∧ ∃𝑥(𝜑𝜒)) → ∃𝑥(𝜑𝜓𝜒))

Proof of Theorem mopick2
StepHypRef Expression
1 hbmo1 2037 . . . 4 (∃*𝑥𝜑 → ∀𝑥∃*𝑥𝜑)
2 hbe1 1471 . . . 4 (∃𝑥(𝜑𝜓) → ∀𝑥𝑥(𝜑𝜓))
31, 2hban 1526 . . 3 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)))
4 mopick 2077 . . . . . 6 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
54ancld 323 . . . . 5 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑 → (𝜑𝜓)))
65anim1d 334 . . . 4 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ((𝜑𝜒) → ((𝜑𝜓) ∧ 𝜒)))
7 df-3an 964 . . . 4 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
86, 7syl6ibr 161 . . 3 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ((𝜑𝜒) → (𝜑𝜓𝜒)))
93, 8eximdh 1590 . 2 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (∃𝑥(𝜑𝜒) → ∃𝑥(𝜑𝜓𝜒)))
1093impia 1178 1 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓) ∧ ∃𝑥(𝜑𝜒)) → ∃𝑥(𝜑𝜓𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 962  ∃wex 1468  ∃*wmo 2000 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515 This theorem depends on definitions:  df-bi 116  df-3an 964  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003 This theorem is referenced by: (None)
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