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Theorem axext4 2420
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 2417 and df-cleq 2429. (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 1730 . . 3  |-  ( x  =  y  ->  (
z  e.  x  <->  z  e.  y ) )
21alrimiv 1641 . 2  |-  ( x  =  y  ->  A. z
( z  e.  x  <->  z  e.  y ) )
3 axext3 2419 . 2  |-  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
42, 3impbii 181 1  |-  ( x  =  y  <->  A. z
( z  e.  x  <->  z  e.  y ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   A.wal 1549
This theorem is referenced by:  ax10ext  27583
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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