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Theorem mo2icl 3228
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 2472 . . . . . 6  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
21imbi2d 322 . . . . 5  |-  ( y  =  A  ->  (
( ph  ->  x  =  y )  <->  ( ph  ->  x  =  A ) ) )
32albidv 1777 . . . 4  |-  ( y  =  A  ->  ( A. x ( ph  ->  x  =  y )  <->  A. x
( ph  ->  x  =  A ) ) )
43imbi1d 323 . . 3  |-  ( y  =  A  ->  (
( A. x (
ph  ->  x  =  y )  ->  E* x ph )  <->  ( A. x
( ph  ->  x  =  A )  ->  E* x ph ) ) )
5 19.8a 1945 . . . 4  |-  ( A. x ( ph  ->  x  =  y )  ->  E. y A. x (
ph  ->  x  =  y ) )
6 mo2v 2316 . . . 4  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
75, 6sylibr 217 . . 3  |-  ( A. x ( ph  ->  x  =  y )  ->  E* x ph )
84, 7vtoclg 3118 . 2  |-  ( A  e.  _V  ->  ( A. x ( ph  ->  x  =  A )  ->  E* x ph ) )
9 eqvisset 3064 . . . . . 6  |-  ( x  =  A  ->  A  e.  _V )
109imim2i 16 . . . . 5  |-  ( (
ph  ->  x  =  A )  ->  ( ph  ->  A  e.  _V )
)
1110con3rr3 143 . . . 4  |-  ( -.  A  e.  _V  ->  ( ( ph  ->  x  =  A )  ->  -.  ph ) )
1211alimdv 1773 . . 3  |-  ( -.  A  e.  _V  ->  ( A. x ( ph  ->  x  =  A )  ->  A. x  -.  ph ) )
13 alnex 1675 . . . 4  |-  ( A. x  -.  ph  <->  -.  E. x ph )
14 exmo 2334 . . . . 5  |-  ( E. x ph  \/  E* x ph )
1514ori 381 . . . 4  |-  ( -. 
E. x ph  ->  E* x ph )
1613, 15sylbi 200 . . 3  |-  ( A. x  -.  ph  ->  E* x ph )
1712, 16syl6 34 . 2  |-  ( -.  A  e.  _V  ->  ( A. x ( ph  ->  x  =  A )  ->  E* x ph ) )
188, 17pm2.61i 169 1  |-  ( A. x ( ph  ->  x  =  A )  ->  E* x ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1452    = wceq 1454   E.wex 1673    e. wcel 1897   E*wmo 2310   _Vcvv 3056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-v 3058
This theorem is referenced by:  invdisj  4404  opabiotafun  5948  fseqenlem2  8481  dfac2  8586  imasaddfnlem  15482  imasvscafn  15491  bnj149  29734
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