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Theorem axext3 2279
Description: A generalization of the Axiom of Extensionality in which  x and  y need not be distinct. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
Assertion
Ref Expression
axext3  |-  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
Distinct variable groups:    x, z    y, z

Proof of Theorem axext3
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 elequ2 1701 . . . . 5  |-  ( w  =  x  ->  (
z  e.  w  <->  z  e.  x ) )
21bibi1d 310 . . . 4  |-  ( w  =  x  ->  (
( z  e.  w  <->  z  e.  y )  <->  ( z  e.  x  <->  z  e.  y ) ) )
32albidv 1615 . . 3  |-  ( w  =  x  ->  ( A. z ( z  e.  w  <->  z  e.  y )  <->  A. z ( z  e.  x  <->  z  e.  y ) ) )
4 equequ1 1667 . . 3  |-  ( w  =  x  ->  (
w  =  y  <->  x  =  y ) )
53, 4imbi12d 311 . 2  |-  ( w  =  x  ->  (
( A. z ( z  e.  w  <->  z  e.  y )  ->  w  =  y )  <->  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y ) ) )
6 ax-ext 2277 . 2  |-  ( A. z ( z  e.  w  <->  z  e.  y )  ->  w  =  y )
75, 6chvarv 1966 1  |-  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530    = wceq 1632    e. wcel 1696
This theorem is referenced by:  axext4  2280  axextnd  8229  axextdist  24227
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535
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