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Theorem mopick 2657
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 2565 . . 3 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 sp 2215 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
3 pm3.45 615 . . . . . . 7 ((𝜑𝑥 = 𝑦) → ((𝜑𝜓) → (𝑥 = 𝑦𝜓)))
43aleximi 1926 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → (∃𝑥(𝜑𝜓) → ∃𝑥(𝑥 = 𝑦𝜓)))
5 sb56 2282 . . . . . . 7 (∃𝑥(𝑥 = 𝑦𝜓) ↔ ∀𝑥(𝑥 = 𝑦𝜓))
6 sp 2215 . . . . . . 7 (∀𝑥(𝑥 = 𝑦𝜓) → (𝑥 = 𝑦𝜓))
75, 6sylbi 208 . . . . . 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 2025 . . 3 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → (∃𝑥(𝜑𝜓) → (𝜑𝜓)))
111, 10sylbi 208 . 2 (∃*𝑥𝜑 → (∃𝑥(𝜑𝜓) → (𝜑𝜓)))
1211imp 395 1 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1650  wex 1874  ∃*wmo 2563
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 2070  ax-7 2105  ax-10 2183  ax-12 2211
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-ex 1875  df-nf 1879  df-mo 2565
This theorem is referenced by:  eupick  2658  mopick2  2662  moexex  2663  morex  3549  imadif  6151  metsscmetcld  23392
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