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Theorem mo2icl 3245
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 316 . . . . 5  |-  ( y  =  A  ->  (
( ph  ->  x  =  y )  <->  ( ph  ->  x  =  A ) ) )
32albidv 1680 . . . 4  |-  ( y  =  A  ->  ( A. x ( ph  ->  x  =  y )  <->  A. x
( ph  ->  x  =  A ) ) )
43imbi1d 317 . . 3  |-  ( y  =  A  ->  (
( A. x (
ph  ->  x  =  y )  ->  E* x ph )  <->  ( A. x
( ph  ->  x  =  A )  ->  E* x ph ) ) )
5 19.8a 1797 . . . 4  |-  ( A. x ( ph  ->  x  =  y )  ->  E. y A. x (
ph  ->  x  =  y ) )
6 mo2v 2269 . . . 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 3136 . 2  |-  ( A  e.  _V  ->  ( A. x ( ph  ->  x  =  A )  ->  E* x ph ) )
9 eqvisset 3086 . . . . . 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 1676 . . 3  |-  ( -.  A  e.  _V  ->  ( A. x ( ph  ->  x  =  A )  ->  A. x  -.  ph ) )
13 alnex 1589 . . . 4  |-  ( A. x  -.  ph  <->  -.  E. x ph )
14 exmo 2291 . . . . 5  |-  ( E. x ph  \/  E* x ph )
1514ori 375 . . . 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 1368    = wceq 1370   E.wex 1587    e. wcel 1758   E*wmo 2263   _Vcvv 3078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-v 3080
This theorem is referenced by:  invdisj  4392  opabiotafun  5864  fseqenlem2  8309  dfac2  8414  imasaddfnlem  14588  imasvscafn  14597  bnj149  32220
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