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Theorem euf 1953
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  |-  ( ph  ->  A. 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 1951 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 euf.1 . . . . 5  |-  ( ph  ->  A. y ph )
3 ax-17 1464 . . . . 5  |-  ( x  =  z  ->  A. y  x  =  z )
42, 3hbbi 1485 . . . 4  |-  ( (
ph 
<->  x  =  z )  ->  A. y ( ph  <->  x  =  z ) )
54hbal 1411 . . 3  |-  ( A. x ( ph  <->  x  =  z )  ->  A. y A. x ( ph  <->  x  =  z ) )
6 ax-17 1464 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  A. z A. x ( ph  <->  x  =  y ) )
7 equequ2 1646 . . . . 5  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
87bibi2d 230 . . . 4  |-  ( z  =  y  ->  (
( ph  <->  x  =  z
)  <->  ( ph  <->  x  =  y ) ) )
98albidv 1752 . . 3  |-  ( z  =  y  ->  ( A. x ( ph  <->  x  =  z )  <->  A. x
( ph  <->  x  =  y
) ) )
105, 6, 9cbvexh 1685 . 2  |-  ( E. z A. x (
ph 
<->  x  =  z )  <->  E. y A. x (
ph 
<->  x  =  y ) )
111, 10bitri 182 1  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287   E.wex 1426   E!weu 1948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-eu 1951
This theorem is referenced by:  eu1  1973  eumo0  1979
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