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Theorem mo3h 1935
Description: Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that y not occur in φ in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) (New usage is discouraged.)
Hypothesis
Ref Expression
mo3h.1 (φyφ)
Assertion
Ref Expression
mo3h (∃*xφxy((φ [y / x]φ) → x = y))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem mo3h
StepHypRef Expression
1 mo3h.1 . . . . . . 7 (φyφ)
21nfi 1331 . . . . . 6 yφ
32eu2 1926 . . . . 5 (∃!xφ ↔ (xφ xy((φ [y / x]φ) → x = y)))
43imbi2i 215 . . . 4 ((xφ∃!xφ) ↔ (xφ → (xφ xy((φ [y / x]φ) → x = y))))
5 df-mo 1886 . . . 4 (∃*xφ ↔ (xφ∃!xφ))
6 anclb 302 . . . 4 ((xφxy((φ [y / x]φ) → x = y)) ↔ (xφ → (xφ xy((φ [y / x]φ) → x = y))))
74, 5, 63bitr4i 201 . . 3 (∃*xφ ↔ (xφxy((φ [y / x]φ) → x = y)))
8 19.38 1548 . . . . 5 ((xφxy((φ [y / x]φ) → x = y)) → x(φy((φ [y / x]φ) → x = y)))
9219.21 1457 . . . . . 6 (y(φ → ((φ [y / x]φ) → x = y)) ↔ (φy((φ [y / x]φ) → x = y)))
109albii 1339 . . . . 5 (xy(φ → ((φ [y / x]φ) → x = y)) ↔ x(φy((φ [y / x]φ) → x = y)))
118, 10sylibr 137 . . . 4 ((xφxy((φ [y / x]φ) → x = y)) → xy(φ → ((φ [y / x]φ) → x = y)))
12 anabs5 494 . . . . . 6 ((φ (φ [y / x]φ)) ↔ (φ [y / x]φ))
13 pm3.31 249 . . . . . 6 ((φ → ((φ [y / x]φ) → x = y)) → ((φ (φ [y / x]φ)) → x = y))
1412, 13syl5bir 142 . . . . 5 ((φ → ((φ [y / x]φ) → x = y)) → ((φ [y / x]φ) → x = y))
15142alimi 1325 . . . 4 (xy(φ → ((φ [y / x]φ) → x = y)) → xy((φ [y / x]φ) → x = y))
1611, 15syl 14 . . 3 ((xφxy((φ [y / x]φ) → x = y)) → xy((φ [y / x]φ) → x = y))
177, 16sylbi 114 . 2 (∃*xφxy((φ [y / x]φ) → x = y))
183simplbi2com 1312 . . 3 (xy((φ [y / x]φ) → x = y) → (xφ∃!xφ))
1918, 5sylibr 137 . 2 (xy((φ [y / x]φ) → x = y) → ∃*xφ)
2017, 19impbii 117 1 (∃*xφxy((φ [y / x]φ) → x = y))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1226  wex 1362  [wsb 1627  ∃!weu 1882  ∃*wmo 1883
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886
This theorem is referenced by:  mo3  1936  mo2dc  1937  mo4f  1942  moim  1946  moimv  1948  moanim  1956  mopick  1960
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