ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  axext3 Unicode version

Theorem axext3 2065
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 1642 . . . . 5  |-  ( w  =  x  ->  (
z  e.  w  <->  z  e.  x ) )
21bibi1d 231 . . . 4  |-  ( w  =  x  ->  (
( z  e.  w  <->  z  e.  y )  <->  ( z  e.  x  <->  z  e.  y ) ) )
32albidv 1746 . . 3  |-  ( w  =  x  ->  ( A. z ( z  e.  w  <->  z  e.  y )  <->  A. z ( z  e.  x  <->  z  e.  y ) ) )
4 equequ1 1639 . . 3  |-  ( w  =  x  ->  (
w  =  y  <->  x  =  y ) )
53, 4imbi12d 232 . 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 2064 . 2  |-  ( A. z ( z  e.  w  <->  z  e.  y )  ->  w  =  y )
75, 6chvarv 1854 1  |-  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  axext4  2066
  Copyright terms: Public domain W3C validator