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Theorem axextdfeq 24156
Description: A version of ax-ext 2266 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 8215 . . 3  |-  E. z
( ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
2 ax-13 1688 . . . . 5  |-  ( x  =  y  ->  (
x  e.  w  -> 
y  e.  w ) )
32imim2i 13 . . . 4  |-  ( ( ( z  e.  x  <->  z  e.  y )  ->  x  =  y )  ->  ( ( z  e.  x  <->  z  e.  y )  ->  ( x  e.  w  ->  y  e.  w ) ) )
43eximi 1565 . . 3  |-  ( E. z ( ( z  e.  x  <->  z  e.  y )  ->  x  =  y )  ->  E. z ( ( z  e.  x  <->  z  e.  y )  ->  (
x  e.  w  -> 
y  e.  w ) ) )
51, 4ax-mp 8 . 2  |-  E. z
( ( z  e.  x  <->  z  e.  y )  ->  ( x  e.  w  ->  y  e.  w ) )
6 biimpexp 24072 . . 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 ) ) ) )
76exbii 1571 . 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 ) ) ) )
85, 7mpbi 199 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 176   E.wex 1530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-cleq 2278  df-clel 2281  df-nfc 2410
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