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Theorem axext2 2265
Description: The Axiom of Extensionality (ax-ext 2264) 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 2264 . 2  |-  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
2 19.36v 1837 . 2  |-  ( E. z ( ( z  e.  x  <->  z  e.  y )  ->  x  =  y )  <->  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y ) )
31, 2mpbir 200 1  |-  E. z
( ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527   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-6 1703  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532
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