MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mo2icl Structured version   Unicode version

Theorem mo2icl 3115
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 2447 . . . . . 6  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
21imbi2d 309 . . . . 5  |-  ( y  =  A  ->  (
( ph  ->  x  =  y )  <->  ( ph  ->  x  =  A ) ) )
32albidv 1636 . . . 4  |-  ( y  =  A  ->  ( A. x ( ph  ->  x  =  y )  <->  A. x
( ph  ->  x  =  A ) ) )
43imbi1d 310 . . 3  |-  ( y  =  A  ->  (
( A. x (
ph  ->  x  =  y )  ->  E* x ph )  <->  ( A. x
( ph  ->  x  =  A )  ->  E* x ph ) ) )
5 19.8a 1763 . . . 4  |-  ( A. x ( ph  ->  x  =  y )  ->  E. y A. x (
ph  ->  x  =  y ) )
6 nfv 1630 . . . . 5  |-  F/ y
ph
76mo2 2312 . . . 4  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
85, 7sylibr 205 . . 3  |-  ( A. x ( ph  ->  x  =  y )  ->  E* x ph )
94, 8vtoclg 3013 . 2  |-  ( A  e.  _V  ->  ( A. x ( ph  ->  x  =  A )  ->  E* x ph ) )
10 vex 2961 . . . . . . 7  |-  x  e. 
_V
11 eleq1 2498 . . . . . . 7  |-  ( x  =  A  ->  (
x  e.  _V  <->  A  e.  _V ) )
1210, 11mpbii 204 . . . . . 6  |-  ( x  =  A  ->  A  e.  _V )
1312imim2i 14 . . . . 5  |-  ( (
ph  ->  x  =  A )  ->  ( ph  ->  A  e.  _V )
)
1413con3rr3 131 . . . 4  |-  ( -.  A  e.  _V  ->  ( ( ph  ->  x  =  A )  ->  -.  ph ) )
1514alimdv 1632 . . 3  |-  ( -.  A  e.  _V  ->  ( A. x ( ph  ->  x  =  A )  ->  A. x  -.  ph ) )
16 alnex 1553 . . . 4  |-  ( A. x  -.  ph  <->  -.  E. x ph )
17 exmo 2328 . . . . 5  |-  ( E. x ph  \/  E* x ph )
1817ori 366 . . . 4  |-  ( -. 
E. x ph  ->  E* x ph )
1916, 18sylbi 189 . . 3  |-  ( A. x  -.  ph  ->  E* x ph )
2015, 19syl6 32 . 2  |-  ( -.  A  e.  _V  ->  ( A. x ( ph  ->  x  =  A )  ->  E* x ph ) )
219, 20pm2.61i 159 1  |-  ( A. x ( ph  ->  x  =  A )  ->  E* x ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1726   E*wmo 2284   _Vcvv 2958
This theorem is referenced by:  invdisj  4204  opabiotafun  6539  fseqenlem2  7911  dfac2  8016  imasaddfnlem  13758  imasvscafn  13767  bnj149  29320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960
  Copyright terms: Public domain W3C validator