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Theorem mopick2 1997
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 1536. (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 1952 . . . 4 (∃*𝑥𝜑 → ∀𝑥∃*𝑥𝜑)
2 hbe1 1398 . . . 4 (∃𝑥(𝜑𝜓) → ∀𝑥𝑥(𝜑𝜓))
31, 2hban 1453 . . 3 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)))
4 mopick 1992 . . . . . 6 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
54ancld 312 . . . . 5 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑 → (𝜑𝜓)))
65anim1d 323 . . . 4 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ((𝜑𝜒) → ((𝜑𝜓) ∧ 𝜒)))
7 df-3an 896 . . . 4 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
86, 7syl6ibr 155 . . 3 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ((𝜑𝜒) → (𝜑𝜓𝜒)))
93, 8eximdh 1516 . 2 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (∃𝑥(𝜑𝜒) → ∃𝑥(𝜑𝜓𝜒)))
1093impia 1110 1 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓) ∧ ∃𝑥(𝜑𝜒)) → ∃𝑥(𝜑𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  w3a 894  wex 1395  ∃*wmo 1915
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442
This theorem depends on definitions:  df-bi 114  df-3an 896  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918
This theorem is referenced by: (None)
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