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Theorem euf 2149
Description: A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.)
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 2147 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 euf.1 . . . . 5  |-  F/ y
ph
3 nfv 1605 . . . . 5  |-  F/ y  x  =  z
42, 3nfbi 1772 . . . 4  |-  F/ y ( ph  <->  x  =  z )
54nfal 1766 . . 3  |-  F/ y A. x ( ph  <->  x  =  z )
6 nfv 1605 . . . . 5  |-  F/ z
ph
7 nfv 1605 . . . . 5  |-  F/ z  x  =  y
86, 7nfbi 1772 . . . 4  |-  F/ z ( ph  <->  x  =  y )
98nfal 1766 . . 3  |-  F/ z A. x ( ph  <->  x  =  y )
10 equequ2 1649 . . . . 5  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
1110bibi2d 309 . . . 4  |-  ( z  =  y  ->  (
( ph  <->  x  =  z
)  <->  ( ph  <->  x  =  y ) ) )
1211albidv 1611 . . 3  |-  ( z  =  y  ->  ( A. x ( ph  <->  x  =  z )  <->  A. x
( ph  <->  x  =  y
) ) )
135, 9, 12cbvex 1925 . 2  |-  ( E. z A. x (
ph 
<->  x  =  z )  <->  E. y A. x (
ph 
<->  x  =  y ) )
141, 13bitri 240 1  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1527   E.wex 1528   F/wnf 1531    = wceq 1623   E!weu 2143
This theorem is referenced by:  eu1  2164  eumo0  2167
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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