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Theorem cleljustALT 2083
Description: Alternate proof of cleljust 2082. (Contributed by NM, 28-Jan-2004.) (Revised by Mario Carneiro, 21-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
cleljustALT  |-  ( x  e.  y  <->  E. z
( z  =  x  /\  z  e.  y ) )
Distinct variable groups:    x, z    y, z

Proof of Theorem cleljustALT
StepHypRef Expression
1 nfv 1683 . . 3  |-  F/ z  x  e.  y
2 elequ1 1770 . . 3  |-  ( z  =  x  ->  (
z  e.  y  <->  x  e.  y ) )
31, 2equsex 2011 . 2  |-  ( E. z ( z  =  x  /\  z  e.  y )  <->  x  e.  y )
43bicomi 202 1  |-  ( x  e.  y  <->  E. z
( z  =  x  /\  z  e.  y ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369   E.wex 1596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-10 1786  ax-12 1803  ax-13 1968
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600
This theorem is referenced by: (None)
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