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Theorem mopick2 2569
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 1837. (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 nfmo1 2509 . . . 4 𝑥∃*𝑥𝜑
2 nfe1 2067 . . . 4 𝑥𝑥(𝜑𝜓)
31, 2nfan 1868 . . 3 𝑥(∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓))
4 mopick 2564 . . . . . 6 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
54ancld 575 . . . . 5 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑 → (𝜑𝜓)))
65anim1d 587 . . . 4 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ((𝜑𝜒) → ((𝜑𝜓) ∧ 𝜒)))
7 df-3an 1056 . . . 4 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
86, 7syl6ibr 242 . . 3 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ((𝜑𝜒) → (𝜑𝜓𝜒)))
93, 8eximd 2123 . 2 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (∃𝑥(𝜑𝜒) → ∃𝑥(𝜑𝜓𝜒)))
1093impia 1280 1 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓) ∧ ∃𝑥(𝜑𝜒)) → ∃𝑥(𝜑𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054  wex 1744  ∃*wmo 2499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-eu 2502  df-mo 2503
This theorem is referenced by:  moantr  34267
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