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Theorem bj-eumo0 31511
Description: Existential uniqueness implies "at most one." Used to be in the main part and deprecated in favor of eumo 2348 and mo2 2328. (Contributed by NM, 8-Jul-1994.) (Revised by BJ, 8-Jun-2019.)
Hypothesis
Ref Expression
bj-eumo0.1  |-  F/ y
ph
Assertion
Ref Expression
bj-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 bj-eumo0
StepHypRef Expression
1 bj-eumo0.1 . . 3  |-  F/ y
ph
21euf 2327 . 2  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
3 biimp 198 . . . 4  |-  ( (
ph 
<->  x  =  y )  ->  ( ph  ->  x  =  y ) )
43alimi 1692 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  A. x
( ph  ->  x  =  y ) )
54eximi 1715 . 2  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  E. y A. x
( ph  ->  x  =  y ) )
62, 5sylbi 200 1  |-  ( E! x ph  ->  E. y A. x ( ph  ->  x  =  y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1450   E.wex 1671   F/wnf 1675   E!weu 2319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-eu 2323
This theorem is referenced by: (None)
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