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Theorem mopick 2266
 Description: "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.)
Assertion
Ref Expression
mopick ((∃*xφ x(φ ψ)) → (φψ))

Proof of Theorem mopick
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . . 4 y(φ ψ)
2 nfs1v 2106 . . . . 5 x[y / x]φ
3 nfs1v 2106 . . . . 5 x[y / x]ψ
42, 3nfan 1824 . . . 4 x([y / x]φ [y / x]ψ)
5 sbequ12 1919 . . . . 5 (x = y → (φ ↔ [y / x]φ))
6 sbequ12 1919 . . . . 5 (x = y → (ψ ↔ [y / x]ψ))
75, 6anbi12d 691 . . . 4 (x = y → ((φ ψ) ↔ ([y / x]φ [y / x]ψ)))
81, 4, 7cbvex 1985 . . 3 (x(φ ψ) ↔ y([y / x]φ [y / x]ψ))
9 nfv 1619 . . . . . . 7 yφ
109mo3 2235 . . . . . 6 (∃*xφxy((φ [y / x]φ) → x = y))
11 sp 1747 . . . . . . 7 (y((φ [y / x]φ) → x = y) → ((φ [y / x]φ) → x = y))
1211sps 1754 . . . . . 6 (xy((φ [y / x]φ) → x = y) → ((φ [y / x]φ) → x = y))
1310, 12sylbi 187 . . . . 5 (∃*xφ → ((φ [y / x]φ) → x = y))
14 sbequ2 1650 . . . . . . . . 9 (x = y → ([y / x]ψψ))
1514imim2i 13 . . . . . . . 8 (((φ [y / x]φ) → x = y) → ((φ [y / x]φ) → ([y / x]ψψ)))
1615exp3a 425 . . . . . . 7 (((φ [y / x]φ) → x = y) → (φ → ([y / x]φ → ([y / x]ψψ))))
1716com4t 79 . . . . . 6 ([y / x]φ → ([y / x]ψ → (((φ [y / x]φ) → x = y) → (φψ))))
1817imp 418 . . . . 5 (([y / x]φ [y / x]ψ) → (((φ [y / x]φ) → x = y) → (φψ)))
1913, 18syl5 28 . . . 4 (([y / x]φ [y / x]ψ) → (∃*xφ → (φψ)))
2019exlimiv 1634 . . 3 (y([y / x]φ [y / x]ψ) → (∃*xφ → (φψ)))
218, 20sylbi 187 . 2 (x(φ ψ) → (∃*xφ → (φψ)))
2221impcom 419 1 ((∃*xφ x(φ ψ)) → (φψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642  [wsb 1648  ∃*wmo 2205 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209 This theorem is referenced by:  eupick  2267  mopick2  2271  moexex  2273  morex  3020  imadif  5171
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