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Theorem mo3 2356
 Description: Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that not occur in in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Aug-2019.)
Hypothesis
Ref Expression
mo3.1
Assertion
Ref Expression
mo3
Distinct variable group:   ,
Allowed substitution hints:   (,)

Proof of Theorem mo3
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfmo1 2330 . . 3
2 mo3.1 . . . . 5
32nfmo 2336 . . . 4
4 mo2v 2326 . . . . 5
5 sp 1957 . . . . . . . 8
6 spsbim 2243 . . . . . . . . 9
7 equsb3 2281 . . . . . . . . 9
86, 7syl6ib 234 . . . . . . . 8
95, 8anim12d 572 . . . . . . 7
10 equtr2 1877 . . . . . . 7
119, 10syl6 33 . . . . . 6
1211exlimiv 1784 . . . . 5
134, 12sylbi 200 . . . 4
143, 13alrimi 1975 . . 3
151, 14alrimi 1975 . 2
16 nfs1v 2286 . . . . . . . 8
17 pm3.21 455 . . . . . . . . 9
1817imim1d 77 . . . . . . . 8
1916, 18alimd 1974 . . . . . . 7
2019com12 31 . . . . . 6
2120aleximi 1712 . . . . 5
222sb8e 2274 . . . . 5
232mo2 2328 . . . . 5
2421, 22, 233imtr4g 278 . . . 4
25 moabs 2350 . . . 4
2624, 25sylibr 217 . . 3
2726alcoms 1938 . 2
2815, 27impbii 192 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376  wal 1450  wex 1671  wnf 1675  wsb 1805  wmo 2320 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324 This theorem is referenced by:  mo  2357  eu2  2358  mo4f  2365  2mo  2400  rmo3  3344  isarep2  5673  mo5f  28199  rmo3f  28210  rmo4fOLD  28211  bnj580  29796  pm14.12  36842
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