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Theorem bj-cleljust 30908
Description: Remove dependency on ax-13 2026 from cleljust 2133. (Contributed by BJ, 27-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-cleljust  |-  ( x  e.  y  <->  E. z
( z  =  x  /\  z  e.  y ) )
Distinct variable groups:    x, z    y, z

Proof of Theorem bj-cleljust
StepHypRef Expression
1 ax-5 1725 . . 3  |-  ( x  e.  y  ->  A. z  x  e.  y )
2 elequ1 1845 . . 3  |-  ( z  =  x  ->  (
z  e.  y  <->  x  e.  y ) )
31, 2bj-equsexhv 30878 . 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 367   E.wex 1633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-10 1861  ax-12 1878
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1634  df-nf 1638
This theorem is referenced by:  bj-dfclel  31027
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