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Theorem mo 2309
 Description: Equivalent definitions of "there exists at most one." (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypothesis
Ref Expression
mo.1
Assertion
Ref Expression
mo
Distinct variable group:   ,
Allowed substitution hints:   (,)

Proof of Theorem mo
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 mo.1 . . . . . 6
2 nfv 1630 . . . . . 6
31, 2nfim 1834 . . . . 5
43nfal 1866 . . . 4
5 nfv 1630 . . . 4
6 equequ2 1700 . . . . . 6
76imbi2d 309 . . . . 5
87albidv 1636 . . . 4
94, 5, 8cbvex 1986 . . 3
10 nfs1v 2188 . . . . . . . . 9
11 nfv 1630 . . . . . . . . 9
1210, 11nfim 1834 . . . . . . . 8
13 sbequ2 1661 . . . . . . . . 9
14 ax-8 1689 . . . . . . . . 9
1513, 14imim12d 71 . . . . . . . 8
163, 12, 15cbv3 1974 . . . . . . 7
1716ancli 536 . . . . . 6
183, 12aaan 1909 . . . . . 6
1917, 18sylibr 205 . . . . 5
20 prth 556 . . . . . . 7
21 equtr2 1702 . . . . . . 7
2220, 21syl6 32 . . . . . 6
23222alimi 1570 . . . . 5
2419, 23syl 16 . . . 4
2524exlimiv 1645 . . 3
269, 25sylbir 206 . 2
27 nfa2 1876 . . . 4
28 sp 1765 . . . . . . . 8
2928exp3a 427 . . . . . . 7
3029com3r 76 . . . . . 6
3110, 30alimd 1782 . . . . 5
3231com12 30 . . . 4
3327, 32eximd 1788 . . 3
34 alnex 1553 . . . 4
3510nfn 1813 . . . . . 6
361nfn 1813 . . . . . 6
37 sbequ1 1946 . . . . . . . 8
3837equcoms 1695 . . . . . . 7
3938con3d 128 . . . . . 6
4035, 36, 39cbv3 1974 . . . . 5
41 pm2.21 103 . . . . . 6
4241alimi 1569 . . . . 5
43 19.8a 1764 . . . . 5
4440, 42, 433syl 19 . . . 4
4534, 44sylbir 206 . . 3
4633, 45pm2.61d1 154 . 2
4726, 46impbii 182 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178   wa 360  wal 1550  wex 1551  wnf 1554  wsb 1659 This theorem is referenced by:  eu2  2312  eu3  2313  mo3  2318 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953 This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660
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