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Theorem eu4 2195
Description: Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)
Hypothesis
Ref Expression
eu4.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
eu4  |-  ( E! x ph  <->  ( E. x ph  /\  A. x A. y ( ( ph  /\ 
ps )  ->  x  =  y ) ) )
Distinct variable groups:    x, y    ph, y    ps, x
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem eu4
StepHypRef Expression
1 eu5 2194 . 2  |-  ( E! x ph  <->  ( E. x ph  /\  E* x ph ) )
2 eu4.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
32mo4 2189 . . 3  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
ps )  ->  x  =  y ) )
43anbi2i 675 . 2  |-  ( ( E. x ph  /\  E* x ph )  <->  ( E. x ph  /\  A. x A. y ( ( ph  /\ 
ps )  ->  x  =  y ) ) )
51, 4bitri 240 1  |-  ( E! x ph  <->  ( E. x ph  /\  A. x A. y ( ( ph  /\ 
ps )  ->  x  =  y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531   E!weu 2156   E*wmo 2157
This theorem is referenced by:  euequ1  2244  eueq  2950  euind  2965  uniintsn  3915  eusv1  4544  omeu  6599  eroveu  6769  climeu  12045  pceu  12915  gsumval3eu  15206  unirep  26452  psgneu  27532  rlimdmafv  28145  frgra3vlem2  28425  3vfriswmgralem  28428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161
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