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Theorem mopick 2023
Description: "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.)
Assertion
Ref Expression
mopick ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))

Proof of Theorem mopick
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ax-17 1462 . . . 4 ((𝜑𝜓) → ∀𝑦(𝜑𝜓))
2 hbs1 1859 . . . . 5 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
3 hbs1 1859 . . . . 5 ([𝑦 / 𝑥]𝜓 → ∀𝑥[𝑦 / 𝑥]𝜓)
42, 3hban 1482 . . . 4 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) → ∀𝑥([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
5 sbequ12 1698 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
6 sbequ12 1698 . . . . 5 (𝑥 = 𝑦 → (𝜓 ↔ [𝑦 / 𝑥]𝜓))
75, 6anbi12d 457 . . . 4 (𝑥 = 𝑦 → ((𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓)))
81, 4, 7cbvexh 1682 . . 3 (∃𝑥(𝜑𝜓) ↔ ∃𝑦([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
9 ax-17 1462 . . . . . . 7 (𝜑 → ∀𝑦𝜑)
109mo3h 1998 . . . . . 6 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
11 ax-4 1443 . . . . . . 7 (∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
1211sps 1473 . . . . . 6 (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
1310, 12sylbi 119 . . . . 5 (∃*𝑥𝜑 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
14 sbequ2 1696 . . . . . . . . 9 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜓𝜓))
1514imim2i 12 . . . . . . . 8 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → ([𝑦 / 𝑥]𝜓𝜓)))
1615expd 254 . . . . . . 7 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (𝜑 → ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓𝜓))))
1716com4t 84 . . . . . 6 ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (𝜑𝜓))))
1817imp 122 . . . . 5 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) → (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (𝜑𝜓)))
1913, 18syl5 32 . . . 4 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) → (∃*𝑥𝜑 → (𝜑𝜓)))
2019exlimiv 1532 . . 3 (∃𝑦([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) → (∃*𝑥𝜑 → (𝜑𝜓)))
218, 20sylbi 119 . 2 (∃𝑥(𝜑𝜓) → (∃*𝑥𝜑 → (𝜑𝜓)))
2221impcom 123 1 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1285  wex 1424  [wsb 1689  ∃*wmo 1946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949
This theorem is referenced by:  eupick  2024  mopick2  2028  moexexdc  2029  euexex  2030  morex  2790  imadif  5059  funimaexglem  5062
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