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Theorem eumo0 2167
Description: Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.)
Hypothesis
Ref Expression
eumo0.1  |-  F/ y
ph
Assertion
Ref Expression
eumo0  |-  ( E! x ph  ->  E. y A. x ( ph  ->  x  =  y ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem eumo0
StepHypRef Expression
1 eumo0.1 . . 3  |-  F/ y
ph
21euf 2149 . 2  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
3 bi1 178 . . . 4  |-  ( (
ph 
<->  x  =  y )  ->  ( ph  ->  x  =  y ) )
43alimi 1546 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  A. x
( ph  ->  x  =  y ) )
54eximi 1563 . 2  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  E. y A. x
( ph  ->  x  =  y ) )
62, 5sylbi 187 1  |-  ( E! x ph  ->  E. y A. x ( ph  ->  x  =  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527   E.wex 1528   F/wnf 1531   E!weu 2143
This theorem is referenced by:  eu2  2168  mo2  2172
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-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-eu 2147
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