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Theorem euf 2228
Description: A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2018.)
Hypothesis
Ref Expression
euf.1  |-  F/ y
ph
Assertion
Ref Expression
euf  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem euf
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-eu 2222 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 euf.1 . . . . 5  |-  F/ y
ph
3 nfv 1715 . . . . 5  |-  F/ y  x  =  z
42, 3nfbi 1942 . . . 4  |-  F/ y ( ph  <->  x  =  z )
54nfal 1955 . . 3  |-  F/ y A. x ( ph  <->  x  =  z )
6 nfv 1715 . . 3  |-  F/ z A. x ( ph  <->  x  =  y )
7 equequ2 1807 . . . . 5  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
87bibi2d 316 . . . 4  |-  ( z  =  y  ->  (
( ph  <->  x  =  z
)  <->  ( ph  <->  x  =  y ) ) )
98albidv 1721 . . 3  |-  ( z  =  y  ->  ( A. x ( ph  <->  x  =  z )  <->  A. x
( ph  <->  x  =  y
) ) )
105, 6, 9cbvex 2029 . 2  |-  ( E. z A. x (
ph 
<->  x  =  z )  <->  E. y A. x (
ph 
<->  x  =  y ) )
111, 10bitri 249 1  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   A.wal 1397   E.wex 1620   F/wnf 1624   E!weu 2218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-eu 2222
This theorem is referenced by:  eu1  2262  bj-eumo0  34761
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