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Theorem mo2icl 3020
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 2367 . . . . . 6  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
21imbi2d 307 . . . . 5  |-  ( y  =  A  ->  (
( ph  ->  x  =  y )  <->  ( ph  ->  x  =  A ) ) )
32albidv 1625 . . . 4  |-  ( y  =  A  ->  ( A. x ( ph  ->  x  =  y )  <->  A. x
( ph  ->  x  =  A ) ) )
43imbi1d 308 . . 3  |-  ( y  =  A  ->  (
( A. x (
ph  ->  x  =  y )  ->  E* x ph )  <->  ( A. x
( ph  ->  x  =  A )  ->  E* x ph ) ) )
5 19.8a 1747 . . . 4  |-  ( A. x ( ph  ->  x  =  y )  ->  E. y A. x (
ph  ->  x  =  y ) )
6 nfv 1619 . . . . 5  |-  F/ y
ph
76mo2 2238 . . . 4  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
85, 7sylibr 203 . . 3  |-  ( A. x ( ph  ->  x  =  y )  ->  E* x ph )
94, 8vtoclg 2919 . 2  |-  ( A  e.  _V  ->  ( A. x ( ph  ->  x  =  A )  ->  E* x ph ) )
10 vex 2867 . . . . . . 7  |-  x  e. 
_V
11 eleq1 2418 . . . . . . 7  |-  ( x  =  A  ->  (
x  e.  _V  <->  A  e.  _V ) )
1210, 11mpbii 202 . . . . . 6  |-  ( x  =  A  ->  A  e.  _V )
1312imim2i 13 . . . . 5  |-  ( (
ph  ->  x  =  A )  ->  ( ph  ->  A  e.  _V )
)
1413con3rr3 128 . . . 4  |-  ( -.  A  e.  _V  ->  ( ( ph  ->  x  =  A )  ->  -.  ph ) )
1514alimdv 1621 . . 3  |-  ( -.  A  e.  _V  ->  ( A. x ( ph  ->  x  =  A )  ->  A. x  -.  ph ) )
16 alnex 1543 . . . 4  |-  ( A. x  -.  ph  <->  -.  E. x ph )
17 exmo 2254 . . . . 5  |-  ( E. x ph  \/  E* x ph )
1817ori 364 . . . 4  |-  ( -. 
E. x ph  ->  E* x ph )
1916, 18sylbi 187 . . 3  |-  ( A. x  -.  ph  ->  E* x ph )
2015, 19syl6 29 . 2  |-  ( -.  A  e.  _V  ->  ( A. x ( ph  ->  x  =  A )  ->  E* x ph ) )
219, 20pm2.61i 156 1  |-  ( A. x ( ph  ->  x  =  A )  ->  E* x ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1540   E.wex 1541    = wceq 1642    e. wcel 1710   E*wmo 2210   _Vcvv 2864
This theorem is referenced by:  invdisj  4093  opabiotafun  6378  fseqenlem2  7742  dfac2  7847  imasaddfnlem  13529  imasvscafn  13538  bnj149  28669
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-v 2866
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