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Theorem mopick 2709
 Description: "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.) (Proof shortened by Wolf Lammen, 17-Sep-2019.)
Assertion
Ref Expression
mopick ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))

Proof of Theorem mopick
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-mo 2620 . . 3 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 sp 2174 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
3 pm3.45 621 . . . . . . 7 ((𝜑𝑥 = 𝑦) → ((𝜑𝜓) → (𝑥 = 𝑦𝜓)))
43aleximi 1825 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → (∃𝑥(𝜑𝜓) → ∃𝑥(𝑥 = 𝑦𝜓)))
5 sb56 2271 . . . . . . 7 (∃𝑥(𝑥 = 𝑦𝜓) ↔ ∀𝑥(𝑥 = 𝑦𝜓))
6 sp 2174 . . . . . . 7 (∀𝑥(𝑥 = 𝑦𝜓) → (𝑥 = 𝑦𝜓))
75, 6sylbi 218 . . . . . 6 (∃𝑥(𝑥 = 𝑦𝜓) → (𝑥 = 𝑦𝜓))
84, 7syl6 35 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (∃𝑥(𝜑𝜓) → (𝑥 = 𝑦𝜓)))
92, 8syl5d 73 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (∃𝑥(𝜑𝜓) → (𝜑𝜓)))
109exlimiv 1924 . . 3 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → (∃𝑥(𝜑𝜓) → (𝜑𝜓)))
111, 10sylbi 218 . 2 (∃*𝑥𝜑 → (∃𝑥(𝜑𝜓) → (𝜑𝜓)))
1211imp 407 1 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1528  ∃wex 1773  ∃*wmo 2618 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-12 2169 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620 This theorem is referenced by:  moexexlem  2710  eupick  2717  mopick2  2721  morex  3714  imadif  6435  metsscmetcld  23833
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