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Theorem axext4 2067
Description: A bidirectional version of Extensionality. Although this theorem "looks" like it is just a definition of equality, it requires the Axiom of Extensionality for its proof under our axiomatization. See the comments for ax-ext 2065. (Contributed by NM, 14-Nov-2008.)
Assertion
Ref Expression
axext4  |-  ( x  =  y  <->  A. z
( z  e.  x  <->  z  e.  y ) )
Distinct variable groups:    x, z    y,
z

Proof of Theorem axext4
StepHypRef Expression
1 elequ2 1643 . . 3  |-  ( x  =  y  ->  (
z  e.  x  <->  z  e.  y ) )
21alrimiv 1797 . 2  |-  ( x  =  y  ->  A. z
( z  e.  x  <->  z  e.  y ) )
3 axext3 2066 . 2  |-  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
42, 3impbii 124 1  |-  ( x  =  y  <->  A. z
( z  e.  x  <->  z  e.  y ) )
Colors of variables: wff set class
Syntax hints:    <-> 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 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by: (None)
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