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Theorem axextdfeq 23322
Description: A version of ax-ext 2234 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 8093 . . 3  |-  E. z
( ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
2 ax-13 1625 . . . . 5  |-  ( x  =  y  ->  (
x  e.  w  -> 
y  e.  w ) )
32imim2i 15 . . . 4  |-  ( ( ( z  e.  x  <->  z  e.  y )  ->  x  =  y )  ->  ( ( z  e.  x  <->  z  e.  y )  ->  ( x  e.  w  ->  y  e.  w ) ) )
43eximi 1574 . . 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 10 . 2  |-  E. z
( ( z  e.  x  <->  z  e.  y )  ->  ( x  e.  w  ->  y  e.  w ) )
6 biimpexp 23241 . . 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 1580 . 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 201 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 6    <-> wb 178   E.wex 1537
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-cleq 2246  df-clel 2249  df-nfc 2374
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