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Theorem ax8dfeq 30493
Description: A version of ax-8 1899 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)
Assertion
Ref Expression
ax8dfeq  |-  E. z
( ( z  e.  x  ->  z  e.  y )  ->  (
w  e.  x  ->  w  e.  y )
)

Proof of Theorem ax8dfeq
StepHypRef Expression
1 ax6e 2104 . 2  |-  E. z 
z  =  w
2 ax8 1903 . . . 4  |-  ( w  =  z  ->  (
w  e.  x  -> 
z  e.  x ) )
32equcoms 1874 . . 3  |-  ( z  =  w  ->  (
w  e.  x  -> 
z  e.  x ) )
4 ax8 1903 . . 3  |-  ( z  =  w  ->  (
z  e.  y  ->  w  e.  y )
)
53, 4imim12d 77 . 2  |-  ( z  =  w  ->  (
( z  e.  x  ->  z  e.  y )  ->  ( w  e.  x  ->  w  e.  y ) ) )
61, 5eximii 1719 1  |-  E. z
( ( z  e.  x  ->  z  e.  y )  ->  (
w  e.  x  ->  w  e.  y )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   E.wex 1673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-12 1943  ax-13 2101
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator