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Theorem mo4f 2008
Description: "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)
Hypotheses
Ref Expression
mo4f.1  |-  F/ x ps
mo4f.2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
mo4f  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
ps )  ->  x  =  y ) )
Distinct variable groups:    x, y    ph, y
Allowed substitution hints:    ph( x)    ps( x, y)

Proof of Theorem mo4f
StepHypRef Expression
1 ax-17 1464 . . 3  |-  ( ph  ->  A. y ph )
21mo3h 2001 . 2  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
3 mo4f.1 . . . . . 6  |-  F/ x ps
4 mo4f.2 . . . . . 6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
53, 4sbie 1721 . . . . 5  |-  ( [ y  /  x ] ph 
<->  ps )
65anbi2i 445 . . . 4  |-  ( (
ph  /\  [ y  /  x ] ph )  <->  (
ph  /\  ps )
)
76imbi1i 236 . . 3  |-  ( ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y )  <->  ( ( ph  /\  ps )  ->  x  =  y )
)
872albii 1405 . 2  |-  ( A. x A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  <->  A. x A. y ( ( ph  /\  ps )  ->  x  =  y ) )
92, 8bitri 182 1  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
ps )  ->  x  =  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287   F/wnf 1394   [wsb 1692   E*wmo 1949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952
This theorem is referenced by:  mo4  2009  mob2  2795  moop2  4078  dffun4f  5031
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