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Theorem axext2 2418
Description: The Axiom of Extensionality (ax-ext 2417) restated so that it postulates the existence of a set  z given two arbitrary sets 
x and  y. This way to express it follows the general idea of the other ZFC axioms, which is to postulate the existence of sets given other sets. (Contributed by NM, 28-Sep-2003.)
Assertion
Ref Expression
axext2  |-  E. z
( ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
Distinct variable group:    x, y, z

Proof of Theorem axext2
StepHypRef Expression
1 ax-ext 2417 . 2  |-  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
2 19.36v 1919 . 2  |-  ( E. z ( ( z  e.  x  <->  z  e.  y )  ->  x  =  y )  <->  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y ) )
31, 2mpbir 201 1  |-  E. z
( ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549   E.wex 1550
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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