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Theorem mo2icl 3081
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 2421 . . . . . 6  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
21imbi2d 308 . . . . 5  |-  ( y  =  A  ->  (
( ph  ->  x  =  y )  <->  ( ph  ->  x  =  A ) ) )
32albidv 1632 . . . 4  |-  ( y  =  A  ->  ( A. x ( ph  ->  x  =  y )  <->  A. x
( ph  ->  x  =  A ) ) )
43imbi1d 309 . . 3  |-  ( y  =  A  ->  (
( A. x (
ph  ->  x  =  y )  ->  E* x ph )  <->  ( A. x
( ph  ->  x  =  A )  ->  E* x ph ) ) )
5 19.8a 1758 . . . 4  |-  ( A. x ( ph  ->  x  =  y )  ->  E. y A. x (
ph  ->  x  =  y ) )
6 nfv 1626 . . . . 5  |-  F/ y
ph
76mo2 2291 . . . 4  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
85, 7sylibr 204 . . 3  |-  ( A. x ( ph  ->  x  =  y )  ->  E* x ph )
94, 8vtoclg 2979 . 2  |-  ( A  e.  _V  ->  ( A. x ( ph  ->  x  =  A )  ->  E* x ph ) )
10 vex 2927 . . . . . . 7  |-  x  e. 
_V
11 eleq1 2472 . . . . . . 7  |-  ( x  =  A  ->  (
x  e.  _V  <->  A  e.  _V ) )
1210, 11mpbii 203 . . . . . 6  |-  ( x  =  A  ->  A  e.  _V )
1312imim2i 14 . . . . 5  |-  ( (
ph  ->  x  =  A )  ->  ( ph  ->  A  e.  _V )
)
1413con3rr3 130 . . . 4  |-  ( -.  A  e.  _V  ->  ( ( ph  ->  x  =  A )  ->  -.  ph ) )
1514alimdv 1628 . . 3  |-  ( -.  A  e.  _V  ->  ( A. x ( ph  ->  x  =  A )  ->  A. x  -.  ph ) )
16 alnex 1549 . . . 4  |-  ( A. x  -.  ph  <->  -.  E. x ph )
17 exmo 2307 . . . . 5  |-  ( E. x ph  \/  E* x ph )
1817ori 365 . . . 4  |-  ( -. 
E. x ph  ->  E* x ph )
1916, 18sylbi 188 . . 3  |-  ( A. x  -.  ph  ->  E* x ph )
2015, 19syl6 31 . 2  |-  ( -.  A  e.  _V  ->  ( A. x ( ph  ->  x  =  A )  ->  E* x ph ) )
219, 20pm2.61i 158 1  |-  ( A. x ( ph  ->  x  =  A )  ->  E* x ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1721   E*wmo 2263   _Vcvv 2924
This theorem is referenced by:  invdisj  4169  opabiotafun  6503  fseqenlem2  7870  dfac2  7975  imasaddfnlem  13716  imasvscafn  13725  bnj149  28964
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-v 2926
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