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Theorem mo4f 2059
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 1506 . . 3  |-  ( ph  ->  A. y ph )
21mo3h 2052 . 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 1764 . . . . 5  |-  ( [ y  /  x ] ph 
<->  ps )
65anbi2i 452 . . . 4  |-  ( (
ph  /\  [ y  /  x ] ph )  <->  (
ph  /\  ps )
)
76imbi1i 237 . . 3  |-  ( ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y )  <->  ( ( ph  /\  ps )  ->  x  =  y )
)
872albii 1447 . 2  |-  ( A. x A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  <->  A. x A. y ( ( ph  /\  ps )  ->  x  =  y ) )
92, 8bitri 183 1  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
ps )  ->  x  =  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1329   F/wnf 1436   [wsb 1735   E*wmo 2000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003
This theorem is referenced by:  mo4  2060  mob2  2864  moop2  4173  dffun4f  5139
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