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Theorem axextdfeq 25417
Description: A version of ax-ext 2416 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)
Assertion
Ref Expression
axextdfeq  |-  E. z
( ( z  e.  x  ->  z  e.  y )  ->  (
( z  e.  y  ->  z  e.  x
)  ->  ( x  e.  w  ->  y  e.  w ) ) )

Proof of Theorem axextdfeq
StepHypRef Expression
1 axextnd 8458 . . 3  |-  E. z
( ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
2 ax-13 1727 . . . 4  |-  ( x  =  y  ->  (
x  e.  w  -> 
y  e.  w ) )
32imim2i 14 . . 3  |-  ( ( ( z  e.  x  <->  z  e.  y )  ->  x  =  y )  ->  ( ( z  e.  x  <->  z  e.  y )  ->  ( x  e.  w  ->  y  e.  w ) ) )
41, 3eximii 1587 . 2  |-  E. z
( ( z  e.  x  <->  z  e.  y )  ->  ( x  e.  w  ->  y  e.  w ) )
5 biimpexp 25165 . . 3  |-  ( ( ( z  e.  x  <->  z  e.  y )  -> 
( x  e.  w  ->  y  e.  w ) )  <->  ( ( z  e.  x  ->  z  e.  y )  ->  (
( z  e.  y  ->  z  e.  x
)  ->  ( x  e.  w  ->  y  e.  w ) ) ) )
65exbii 1592 . 2  |-  ( E. z ( ( z  e.  x  <->  z  e.  y )  ->  (
x  e.  w  -> 
y  e.  w ) )  <->  E. z ( ( z  e.  x  -> 
z  e.  y )  ->  ( ( z  e.  y  ->  z  e.  x )  ->  (
x  e.  w  -> 
y  e.  w ) ) ) )
74, 6mpbi 200 1  |-  E. z
( ( z  e.  x  ->  z  e.  y )  ->  (
( z  e.  y  ->  z  e.  x
)  ->  ( x  e.  w  ->  y  e.  w ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   E.wex 1550
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-cleq 2428  df-clel 2431  df-nfc 2560
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