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Theorem axext3 2447
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.) Remove dependencies on ax-10 1786, ax-12 1803, ax-13 1968. (Revised by Wolf Lammen, 9-Dec-2019.)
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 ax6ev 1721 . 2  |-  E. w  w  =  x
2 elequ2 1772 . . . . . . 7  |-  ( w  =  x  ->  (
z  e.  w  <->  z  e.  x ) )
32bibi1d 319 . . . . . 6  |-  ( w  =  x  ->  (
( z  e.  w  <->  z  e.  y )  <->  ( z  e.  x  <->  z  e.  y ) ) )
43albidv 1689 . . . . 5  |-  ( w  =  x  ->  ( A. z ( z  e.  w  <->  z  e.  y )  <->  A. z ( z  e.  x  <->  z  e.  y ) ) )
5 ax-ext 2445 . . . . 5  |-  ( A. z ( z  e.  w  <->  z  e.  y )  ->  w  =  y )
64, 5syl6bir 229 . . . 4  |-  ( w  =  x  ->  ( A. z ( z  e.  x  <->  z  e.  y )  ->  w  =  y ) )
7 ax-7 1739 . . . 4  |-  ( w  =  x  ->  (
w  =  y  ->  x  =  y )
)
86, 7syld 44 . . 3  |-  ( w  =  x  ->  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y ) )
98exlimiv 1698 . 2  |-  ( E. w  w  =  x  ->  ( A. z
( z  e.  x  <->  z  e.  y )  ->  x  =  y )
)
101, 9ax-mp 5 1  |-  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1377   E.wex 1596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-ex 1597
This theorem is referenced by:  axext4  2449  axextnd  8978  axextdist  29159
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