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Theorem cleljust 2164
Description: When the class variables of set theory are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 2159 with the class variables in wcel 2158. (Contributed by NM, 28-Jan-2004.)
Assertion
Ref Expression
cleljust  |-  ( x  e.  y  <->  E. z
( z  =  x  /\  z  e.  y ) )
Distinct variable groups:    x, z    y,
z

Proof of Theorem cleljust
StepHypRef Expression
1 ax-17 1536 . . 3  |-  ( x  e.  y  ->  A. z  x  e.  y )
2 elequ1 2162 . . 3  |-  ( z  =  x  ->  (
z  e.  y  <->  x  e.  y ) )
31, 2equsex 1738 . 2  |-  ( E. z ( z  =  x  /\  z  e.  y )  <->  x  e.  y )
43bicomi 132 1  |-  ( x  e.  y  <->  E. z
( z  =  x  /\  z  e.  y ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105   E.wex 1502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-13 2160
This theorem depends on definitions:  df-bi 117
This theorem is referenced by: (None)
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