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Theorem mopick 1975
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 ax-17 1416 . . . 4 ((φ ψ) → y(φ ψ))
2 hbs1 1811 . . . . 5 ([y / x]φx[y / x]φ)
3 hbs1 1811 . . . . 5 ([y / x]ψx[y / x]ψ)
42, 3hban 1436 . . . 4 (([y / x]φ [y / x]ψ) → x([y / x]φ [y / x]ψ))
5 sbequ12 1651 . . . . 5 (x = y → (φ ↔ [y / x]φ))
6 sbequ12 1651 . . . . 5 (x = y → (ψ ↔ [y / x]ψ))
75, 6anbi12d 442 . . . 4 (x = y → ((φ ψ) ↔ ([y / x]φ [y / x]ψ)))
81, 4, 7cbvexh 1635 . . 3 (x(φ ψ) ↔ y([y / x]φ [y / x]ψ))
9 ax-17 1416 . . . . . . 7 (φyφ)
109mo3h 1950 . . . . . 6 (∃*xφxy((φ [y / x]φ) → x = y))
11 ax-4 1397 . . . . . . 7 (y((φ [y / x]φ) → x = y) → ((φ [y / x]φ) → x = y))
1211sps 1427 . . . . . 6 (xy((φ [y / x]φ) → x = y) → ((φ [y / x]φ) → x = y))
1310, 12sylbi 114 . . . . 5 (∃*xφ → ((φ [y / x]φ) → x = y))
14 sbequ2 1649 . . . . . . . . 9 (x = y → ([y / x]ψψ))
1514imim2i 12 . . . . . . . 8 (((φ [y / x]φ) → x = y) → ((φ [y / x]φ) → ([y / x]ψψ)))
1615expd 245 . . . . . . 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 115 . . . . 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 1486 . . 3 (y([y / x]φ [y / x]ψ) → (∃*xφ → (φψ)))
218, 20sylbi 114 . 2 (x(φ ψ) → (∃*xφ → (φψ)))
2221impcom 116 1 ((∃*xφ x(φ ψ)) → (φψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240  wex 1378  [wsb 1642  ∃*wmo 1898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901
This theorem is referenced by:  eupick  1976  mopick2  1980  moexexdc  1981  euexex  1982  morex  2719  imadif  4922  funimaexglem  4925
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