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Theorem mopick 1994
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 1435 . . . 4 ((𝜑𝜓) → ∀𝑦(𝜑𝜓))
2 hbs1 1830 . . . . 5 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
3 hbs1 1830 . . . . 5 ([𝑦 / 𝑥]𝜓 → ∀𝑥[𝑦 / 𝑥]𝜓)
42, 3hban 1455 . . . 4 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) → ∀𝑥([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
5 sbequ12 1670 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
6 sbequ12 1670 . . . . 5 (𝑥 = 𝑦 → (𝜓 ↔ [𝑦 / 𝑥]𝜓))
75, 6anbi12d 450 . . . 4 (𝑥 = 𝑦 → ((𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓)))
81, 4, 7cbvexh 1654 . . 3 (∃𝑥(𝜑𝜓) ↔ ∃𝑦([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
9 ax-17 1435 . . . . . . 7 (𝜑 → ∀𝑦𝜑)
109mo3h 1969 . . . . . 6 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
11 ax-4 1416 . . . . . . 7 (∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
1211sps 1446 . . . . . 6 (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
1310, 12sylbi 118 . . . . 5 (∃*𝑥𝜑 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
14 sbequ2 1668 . . . . . . . . 9 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜓𝜓))
1514imim2i 12 . . . . . . . 8 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → ([𝑦 / 𝑥]𝜓𝜓)))
1615expd 249 . . . . . . 7 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (𝜑 → ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓𝜓))))
1716com4t 83 . . . . . 6 ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (𝜑𝜓))))
1817imp 119 . . . . 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 1505 . . 3 (∃𝑦([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) → (∃*𝑥𝜑 → (𝜑𝜓)))
218, 20sylbi 118 . 2 (∃𝑥(𝜑𝜓) → (∃*𝑥𝜑 → (𝜑𝜓)))
2221impcom 120 1 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1257  wex 1397  [wsb 1661  ∃*wmo 1917
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 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920
This theorem is referenced by:  eupick  1995  mopick2  1999  moexexdc  2000  euexex  2001  morex  2747  imadif  5006  funimaexglem  5009
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