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Theorem axext3 2419
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 1730 . . . . 5  |-  ( w  =  x  ->  (
z  e.  w  <->  z  e.  x ) )
21bibi1d 311 . . . 4  |-  ( w  =  x  ->  (
( z  e.  w  <->  z  e.  y )  <->  ( z  e.  x  <->  z  e.  y ) ) )
32albidv 1635 . . 3  |-  ( w  =  x  ->  ( A. z ( z  e.  w  <->  z  e.  y )  <->  A. z ( z  e.  x  <->  z  e.  y ) ) )
4 equequ1 1696 . . 3  |-  ( w  =  x  ->  (
w  =  y  <->  x  =  y ) )
53, 4imbi12d 312 . 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 2417 . 2  |-  ( A. z ( z  e.  w  <->  z  e.  y )  ->  w  =  y )
75, 6chvarv 1969 1  |-  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549
This theorem is referenced by:  axext4  2420  axextnd  8466  axextdist  25427
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-14 1729  ax-6 1744  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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