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Theorem axextndbi 24161
Description: axextnd 8213 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 8213 . . 3  |-  E. z
( ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
2 elequ2 1689 . . . . 5  |-  ( x  =  y  ->  (
z  e.  x  <->  z  e.  y ) )
32jctl 525 . . . 4  |-  ( ( ( z  e.  x  <->  z  e.  y )  ->  x  =  y )  ->  ( ( x  =  y  ->  ( z  e.  x  <->  z  e.  y ) )  /\  (
( z  e.  x  <->  z  e.  y )  ->  x  =  y )
) )
43eximi 1563 . . 3  |-  ( E. z ( ( z  e.  x  <->  z  e.  y )  ->  x  =  y )  ->  E. z ( ( x  =  y  ->  (
z  e.  x  <->  z  e.  y ) )  /\  ( ( z  e.  x  <->  z  e.  y )  ->  x  =  y ) ) )
51, 4ax-mp 8 . 2  |-  E. z
( ( x  =  y  ->  ( z  e.  x  <->  z  e.  y ) )  /\  (
( z  e.  x  <->  z  e.  y )  ->  x  =  y )
)
6 dfbi2 609 . . 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 ) ) )
76exbii 1569 . 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 )
) )
85, 7mpbir 200 1  |-  E. z
( x  =  y  <-> 
( z  e.  x  <->  z  e.  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-cleq 2276  df-clel 2279  df-nfc 2408
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