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Theorem mo2icl 3275
Description: Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.)
Assertion
Ref Expression
mo2icl  |-  ( A. x ( ph  ->  x  =  A )  ->  E* x ph )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem mo2icl
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2469 . . . . . 6  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
21imbi2d 314 . . . . 5  |-  ( y  =  A  ->  (
( ph  ->  x  =  y )  <->  ( ph  ->  x  =  A ) ) )
32albidv 1718 . . . 4  |-  ( y  =  A  ->  ( A. x ( ph  ->  x  =  y )  <->  A. x
( ph  ->  x  =  A ) ) )
43imbi1d 315 . . 3  |-  ( y  =  A  ->  (
( A. x (
ph  ->  x  =  y )  ->  E* x ph )  <->  ( A. x
( ph  ->  x  =  A )  ->  E* x ph ) ) )
5 19.8a 1862 . . . 4  |-  ( A. x ( ph  ->  x  =  y )  ->  E. y A. x (
ph  ->  x  =  y ) )
6 mo2v 2291 . . . 4  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
75, 6sylibr 212 . . 3  |-  ( A. x ( ph  ->  x  =  y )  ->  E* x ph )
84, 7vtoclg 3164 . 2  |-  ( A  e.  _V  ->  ( A. x ( ph  ->  x  =  A )  ->  E* x ph ) )
9 eqvisset 3114 . . . . . 6  |-  ( x  =  A  ->  A  e.  _V )
109imim2i 14 . . . . 5  |-  ( (
ph  ->  x  =  A )  ->  ( ph  ->  A  e.  _V )
)
1110con3rr3 136 . . . 4  |-  ( -.  A  e.  _V  ->  ( ( ph  ->  x  =  A )  ->  -.  ph ) )
1211alimdv 1714 . . 3  |-  ( -.  A  e.  _V  ->  ( A. x ( ph  ->  x  =  A )  ->  A. x  -.  ph ) )
13 alnex 1619 . . . 4  |-  ( A. x  -.  ph  <->  -.  E. x ph )
14 exmo 2311 . . . . 5  |-  ( E. x ph  \/  E* x ph )
1514ori 373 . . . 4  |-  ( -. 
E. x ph  ->  E* x ph )
1613, 15sylbi 195 . . 3  |-  ( A. x  -.  ph  ->  E* x ph )
1712, 16syl6 33 . 2  |-  ( -.  A  e.  _V  ->  ( A. x ( ph  ->  x  =  A )  ->  E* x ph ) )
188, 17pm2.61i 164 1  |-  ( A. x ( ph  ->  x  =  A )  ->  E* x ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1396    = wceq 1398   E.wex 1617    e. wcel 1823   E*wmo 2285   _Vcvv 3106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-12 1859  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-v 3108
This theorem is referenced by:  invdisj  4428  opabiotafun  5909  fseqenlem2  8397  dfac2  8502  imasaddfnlem  15017  imasvscafn  15026  bnj149  34334
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