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Theorem axextndbi 25383
Description: axextnd 8430 as a biconditional. (Contributed by Scott Fenton, 14-Dec-2010.)
Assertion
Ref Expression
axextndbi  |-  E. z
( x  =  y  <-> 
( z  e.  x  <->  z  e.  y ) )

Proof of Theorem axextndbi
StepHypRef Expression
1 axextnd 8430 . . 3  |-  E. z
( ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
2 elequ2 1726 . . . 4  |-  ( x  =  y  ->  (
z  e.  x  <->  z  e.  y ) )
32jctl 526 . . 3  |-  ( ( ( z  e.  x  <->  z  e.  y )  ->  x  =  y )  ->  ( ( x  =  y  ->  ( z  e.  x  <->  z  e.  y ) )  /\  (
( z  e.  x  <->  z  e.  y )  ->  x  =  y )
) )
41, 3eximii 1584 . 2  |-  E. z
( ( x  =  y  ->  ( z  e.  x  <->  z  e.  y ) )  /\  (
( z  e.  x  <->  z  e.  y )  ->  x  =  y )
)
5 dfbi2 610 . . 3  |-  ( ( x  =  y  <->  ( z  e.  x  <->  z  e.  y ) )  <->  ( (
x  =  y  -> 
( z  e.  x  <->  z  e.  y ) )  /\  ( ( z  e.  x  <->  z  e.  y )  ->  x  =  y ) ) )
65exbii 1589 . 2  |-  ( E. z ( x  =  y  <->  ( z  e.  x  <->  z  e.  y ) )  <->  E. z
( ( x  =  y  ->  ( z  e.  x  <->  z  e.  y ) )  /\  (
( z  e.  x  <->  z  e.  y )  ->  x  =  y )
) )
74, 6mpbir 201 1  |-  E. z
( x  =  y  <-> 
( z  e.  x  <->  z  e.  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-cleq 2405  df-clel 2408  df-nfc 2537
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