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Theorem eujust 2145
Description: A soundness justification theorem for df-eu 2147, showing that the definition is equivalent to itself with its dummy variable renamed. Note that  y and  z needn't be distinct variables. See eujustALT 2146 for a proof that provides an example of how it can be achieved through the use of dvelim 1956. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
eujust  |-  ( E. y A. x (
ph 
<->  x  =  y )  <->  E. z A. x (
ph 
<->  x  =  z ) )
Distinct variable groups:    x, y    x, z    ph, y    ph, z
Allowed substitution hint:    ph( x)

Proof of Theorem eujust
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 equequ2 1649 . . . . 5  |-  ( y  =  w  ->  (
x  =  y  <->  x  =  w ) )
21bibi2d 309 . . . 4  |-  ( y  =  w  ->  (
( ph  <->  x  =  y
)  <->  ( ph  <->  x  =  w ) ) )
32albidv 1611 . . 3  |-  ( y  =  w  ->  ( A. x ( ph  <->  x  =  y )  <->  A. x
( ph  <->  x  =  w
) ) )
43cbvexv 1943 . 2  |-  ( E. y A. x (
ph 
<->  x  =  y )  <->  E. w A. x (
ph 
<->  x  =  w ) )
5 equequ2 1649 . . . . 5  |-  ( w  =  z  ->  (
x  =  w  <->  x  =  z ) )
65bibi2d 309 . . . 4  |-  ( w  =  z  ->  (
( ph  <->  x  =  w
)  <->  ( ph  <->  x  =  z ) ) )
76albidv 1611 . . 3  |-  ( w  =  z  ->  ( A. x ( ph  <->  x  =  w )  <->  A. x
( ph  <->  x  =  z
) ) )
87cbvexv 1943 . 2  |-  ( E. w A. x (
ph 
<->  x  =  w )  <->  E. z A. x (
ph 
<->  x  =  z ) )
94, 8bitri 240 1  |-  ( E. y A. x (
ph 
<->  x  =  y )  <->  E. z A. x (
ph 
<->  x  =  z ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1527   E.wex 1528    = wceq 1623
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
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