MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axext3 Structured version   Unicode version

Theorem axext3 2425
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 1761 . . . . 5  |-  ( w  =  x  ->  (
z  e.  w  <->  z  e.  x ) )
21bibi1d 319 . . . 4  |-  ( w  =  x  ->  (
( z  e.  w  <->  z  e.  y )  <->  ( z  e.  x  <->  z  e.  y ) ) )
32albidv 1679 . . 3  |-  ( w  =  x  ->  ( A. z ( z  e.  w  <->  z  e.  y )  <->  A. z ( z  e.  x  <->  z  e.  y ) ) )
4 equequ1 1736 . . 3  |-  ( w  =  x  ->  (
w  =  y  <->  x  =  y ) )
53, 4imbi12d 320 . 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 2423 . 2  |-  ( A. z ( z  e.  w  <->  z  e.  y )  ->  w  =  y )
75, 6chvarv 1958 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 1367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1587  df-nf 1590
This theorem is referenced by:  axext4  2426  axextnd  8755  axextdist  27613
  Copyright terms: Public domain W3C validator