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Theorem mopick2 2661
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 1967. (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 2569 . . . 4 𝑥∃*𝑥𝜑
2 nfe1 2192 . . . 4 𝑥𝑥(𝜑𝜓)
31, 2nfan 1998 . . 3 𝑥(∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓))
4 mopick 2656 . . . . . 6 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
54ancld 546 . . . . 5 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑 → (𝜑𝜓)))
65anim1d 604 . . . 4 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ((𝜑𝜒) → ((𝜑𝜓) ∧ 𝜒)))
7 df-3an 1109 . . . 4 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
86, 7syl6ibr 243 . . 3 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ((𝜑𝜒) → (𝜑𝜓𝜒)))
93, 8eximd 2249 . 2 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (∃𝑥(𝜑𝜒) → ∃𝑥(𝜑𝜓𝜒)))
1093impia 1145 1 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓) ∧ ∃𝑥(𝜑𝜒)) → ∃𝑥(𝜑𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107  wex 1874  ∃*wmo 2562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-10 2183  ax-11 2198  ax-12 2211
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-mo 2564
This theorem is referenced by:  moantr  34488
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