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Theorem axext3 2160
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 2153 . . . . 5  |-  ( w  =  x  ->  (
z  e.  w  <->  z  e.  x ) )
21bibi1d 233 . . . 4  |-  ( w  =  x  ->  (
( z  e.  w  <->  z  e.  y )  <->  ( z  e.  x  <->  z  e.  y ) ) )
32albidv 1824 . . 3  |-  ( w  =  x  ->  ( A. z ( z  e.  w  <->  z  e.  y )  <->  A. z ( z  e.  x  <->  z  e.  y ) ) )
4 equequ1 1712 . . 3  |-  ( w  =  x  ->  (
w  =  y  <->  x  =  y ) )
53, 4imbi12d 234 . 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 2159 . 2  |-  ( A. z ( z  e.  w  <->  z  e.  y )  ->  w  =  y )
75, 6chvarv 1937 1  |-  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wal 1351
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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-14 2151  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461
This theorem is referenced by:  axext4  2161
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