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Theorem eu3 2182
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 2181 . 2  |-  ( E! x ph  <->  ( E. x ph  /\  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) ) )
31mo 2178 . . 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 1530   E.wex 1531   F/wnf 1534   [wsb 1638   E!weu 2156
This theorem is referenced by:  mo2  2185  eu5  2194  2eu4  2239  eqeu  2949  reu3  2968  eunex  4219
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
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