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Theorem mo2v 2326
 Description: Alternate definition of "at most one." Unlike mo2 2328, which is slightly more general, it does not depend on ax-11 1937 and ax-13 2104, whence it is preferable within predicate logic. Elsewhere, most theorems depend on these axioms anyway, so this advantage is no longer important. (Contributed by Wolf Lammen, 27-May-2019.) (Proof shortened by Wolf Lammen, 10-Nov-2019.)
Assertion
Ref Expression
mo2v
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem mo2v
StepHypRef Expression
1 df-mo 2324 . 2
2 df-eu 2323 . . 3
32imbi2i 319 . 2
4 alnex 1673 . . . . . . 7
5 pm2.21 111 . . . . . . . 8
65alimi 1692 . . . . . . 7
74, 6sylbir 218 . . . . . 6
87eximi 1715 . . . . 5
9819.23bi 1969 . . . 4
10 biimp 198 . . . . . 6
1110alimi 1692 . . . . 5
1211eximi 1715 . . . 4
139, 12ja 166 . . 3
14 nfia1 2056 . . . . . 6
15 id 22 . . . . . . . . . 10
16 ax12v 1951 . . . . . . . . . . 11
1716com12 31 . . . . . . . . . 10
1815, 17embantd 55 . . . . . . . . 9
1918spsd 1965 . . . . . . . 8
2019ancld 562 . . . . . . 7
21 albiim 1760 . . . . . . 7
2220, 21syl6ibr 235 . . . . . 6
2314, 22exlimi 2015 . . . . 5
2423eximdv 1772 . . . 4
2524com12 31 . . 3
2613, 25impbii 192 . 2
271, 3, 263bitri 279 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 189   wa 376  wal 1450  wex 1671  weu 2319  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-12 1950 This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676  df-eu 2323  df-mo 2324 This theorem is referenced by:  mo2  2328  eu3v  2347  mo3  2356  sbmo  2364  moim  2368  mopick  2384  2mo2  2399  mo2icl  3205  moabex  4659  dffun3  5600  dffun6f  5603  grothprim  9277  bj-mo3OLD  31515  wl-mo2df  31969  wl-mo2t  31974  wl-mo3t  31975  dffrege115  36645
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