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Theorem eu3 2169
Description: An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.)
Hypothesis
Ref Expression
eu3.1  |-  F/ y
ph
Assertion
Ref Expression
eu3  |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem eu3
StepHypRef Expression
1 eu3.1 . . 3  |-  F/ y
ph
21eu2 2168 . 2  |-  ( E! x ph  <->  ( E. x ph  /\  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) ) )
31mo 2165 . . 3  |-  ( E. y A. x (
ph  ->  x  =  y )  <->  A. x A. y
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
43anbi2i 675 . 2  |-  ( ( E. x ph  /\  E. y A. x (
ph  ->  x  =  y ) )  <->  ( E. x ph  /\  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) ) )
52, 4bitr4i 243 1  |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528   F/wnf 1531   [wsb 1629   E!weu 2143
This theorem is referenced by:  mo2  2172  eu5  2181  2eu4  2226  eqeu  2936  reu3  2955  eunex  4203
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147
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