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Theorem axextndbi 25437
Description: axextnd 8471 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 8471 . . 3  |-  E. z
( ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
2 elequ2 1731 . . . 4  |-  ( x  =  y  ->  (
z  e.  x  <->  z  e.  y ) )
32jctl 527 . . 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 1588 . 2  |-  E. z
( ( x  =  y  ->  ( z  e.  x  <->  z  e.  y ) )  /\  (
( z  e.  x  <->  z  e.  y )  ->  x  =  y )
)
5 dfbi2 611 . . 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 1593 . 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 202 1  |-  E. z
( x  =  y  <-> 
( z  e.  x  <->  z  e.  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   E.wex 1551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-cleq 2431  df-clel 2434  df-nfc 2563
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