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Theorem mopick 2625
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 2540 . . 3 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 sp 2183 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
3 pm3.45 622 . . . . . . 7 ((𝜑𝑥 = 𝑦) → ((𝜑𝜓) → (𝑥 = 𝑦𝜓)))
43aleximi 1832 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → (∃𝑥(𝜑𝜓) → ∃𝑥(𝑥 = 𝑦𝜓)))
5 ax12ev2 2180 . . . . . 6 (∃𝑥(𝑥 = 𝑦𝜓) → (𝑥 = 𝑦𝜓))
64, 5syl6 35 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (∃𝑥(𝜑𝜓) → (𝑥 = 𝑦𝜓)))
72, 6syl5d 73 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (∃𝑥(𝜑𝜓) → (𝜑𝜓)))
87exlimiv 1930 . . 3 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → (∃𝑥(𝜑𝜓) → (𝜑𝜓)))
91, 8sylbi 217 . 2 (∃*𝑥𝜑 → (∃𝑥(𝜑𝜓) → (𝜑𝜓)))
109imp 406 1 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1538  wex 1779  ∃*wmo 2538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-12 2177
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-mo 2540
This theorem is referenced by:  moexexlem  2626  eupick  2633  mopick2  2637  morex  3725  imadif  6650  metsscmetcld  25349
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