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Theorem mo 2302
 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 1629 . . . . . 6
31, 2nfim 1832 . . . . 5
43nfal 1864 . . . 4
5 nfv 1629 . . . 4
6 equequ2 1698 . . . . . 6
76imbi2d 308 . . . . 5
87albidv 1635 . . . 4
94, 5, 8cbvex 1983 . . 3
10 nfs1v 2181 . . . . . . . . 9
11 nfv 1629 . . . . . . . . 9
1210, 11nfim 1832 . . . . . . . 8
13 sbequ2 1660 . . . . . . . . 9
14 ax-8 1687 . . . . . . . . 9
1513, 14imim12d 70 . . . . . . . 8
163, 12, 15cbv3 1971 . . . . . . 7
1716ancli 535 . . . . . 6
183, 12aaan 1906 . . . . . 6
1917, 18sylibr 204 . . . . 5
20 prth 555 . . . . . . 7
21 equtr2 1700 . . . . . . 7
2220, 21syl6 31 . . . . . 6
23222alimi 1569 . . . . 5
2419, 23syl 16 . . . 4
2524exlimiv 1644 . . 3
269, 25sylbir 205 . 2
27 nfa2 1874 . . . 4
28 sp 1763 . . . . . . . 8
2928exp3a 426 . . . . . . 7
3029com3r 75 . . . . . 6
3110, 30alimd 1780 . . . . 5
3231com12 29 . . . 4
3327, 32eximd 1786 . . 3
34 alnex 1552 . . . 4
3510nfn 1811 . . . . . 6
361nfn 1811 . . . . . 6
37 sbequ1 1943 . . . . . . . 8
3837equcoms 1693 . . . . . . 7
3938con3d 127 . . . . . 6
4035, 36, 39cbv3 1971 . . . . 5
41 pm2.21 102 . . . . . 6
4241alimi 1568 . . . . 5
43 19.8a 1762 . . . . 5
4440, 42, 433syl 19 . . . 4
4534, 44sylbir 205 . . 3
4633, 45pm2.61d1 153 . 2
4726, 46impbii 181 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wa 359  wal 1549  wex 1550  wnf 1553  wsb 1658 This theorem is referenced by:  eu2  2305  eu3  2306  mo3  2311 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950 This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659
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