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Theorem 2mo 2373
Description: Two equivalent expressions for double "at most one." (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Nov-2019.)
Assertion
Ref Expression
2mo
Distinct variable groups:   , , ,   , ,

Proof of Theorem 2mo
StepHypRef Expression
1 2mo2 2372 . . . 4
2 nfmo1 2295 . . . . . . 7
3 nfe1 1840 . . . . . . . 8
43nfmo 2301 . . . . . . 7
52, 4nfan 1928 . . . . . 6
6 nfe1 1840 . . . . . . . . 9
76nfmo 2301 . . . . . . . 8
8 nfmo1 2295 . . . . . . . 8
97, 8nfan 1928 . . . . . . 7
10 19.8a 1857 . . . . . . . . 9
11 spsbe 1743 . . . . . . . . . 10
1211sbimi 1745 . . . . . . . . 9
13 nfv 1707 . . . . . . . . . . . 12
1413mo3 2323 . . . . . . . . . . 11
1514biimpi 194 . . . . . . . . . 10
161519.21bbi 1870 . . . . . . . . 9
1710, 12, 16syl2ani 656 . . . . . . . 8
18 19.8a 1857 . . . . . . . . 9
19 sbcom2 2189 . . . . . . . . . 10
20 spsbe 1743 . . . . . . . . . . 11
2120sbimi 1745 . . . . . . . . . 10
2219, 21sylbi 195 . . . . . . . . 9
23 nfv 1707 . . . . . . . . . . . 12
2423mo3 2323 . . . . . . . . . . 11
2524biimpi 194 . . . . . . . . . 10
262519.21bbi 1870 . . . . . . . . 9
2718, 22, 26syl2ani 656 . . . . . . . 8
2817, 27anim12ii 570 . . . . . . 7
299, 28alrimi 1877 . . . . . 6
305, 29alrimi 1877 . . . . 5
3130alrimivv 1720 . . . 4
321, 31sylbir 213 . . 3
33 nfs1v 2181 . . . . . . . 8
34 nfs1v 2181 . . . . . . . . . 10
3534nfsb 2184 . . . . . . . . 9
36 pm3.21 448 . . . . . . . . . 10
3736imim1d 75 . . . . . . . . 9
3835, 37alimd 1876 . . . . . . . 8
3933, 38alimd 1876 . . . . . . 7
4039com12 31 . . . . . 6
4140aleximi 1653 . . . . 5
4241aleximi 1653 . . . 4
43 2nexaln 1651 . . . . . 6
44 2sb8e 2211 . . . . . 6
4543, 44xchnxbi 308 . . . . 5
46 pm2.21 108 . . . . . . . . 9
47462alimi 1634 . . . . . . . 8
48472eximi 1657 . . . . . . 7
494819.23bi 1871 . . . . . 6
504919.23bi 1871 . . . . 5
5145, 50sylbi 195 . . . 4
5242, 51pm2.61d1 159 . . 3
5332, 52impbii 188 . 2
54 alrot4 1847 . 2
5553, 54bitri 249 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612  [wsb 1739  E*wmo 2283
This theorem is referenced by:  2mos  2375  2eu6OLD  2384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287
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