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Theorem ax13dfeq 24155
Description: A version of ax-13 1686 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)
Assertion
Ref Expression
ax13dfeq  |-  E. z
( ( z  e.  x  ->  z  e.  y )  ->  (
w  e.  x  ->  w  e.  y )
)

Proof of Theorem ax13dfeq
StepHypRef Expression
1 a9e 1891 . 2  |-  E. z 
z  =  w
2 ax-13 1686 . . . . 5  |-  ( w  =  z  ->  (
w  e.  x  -> 
z  e.  x ) )
32equcoms 1651 . . . 4  |-  ( z  =  w  ->  (
w  e.  x  -> 
z  e.  x ) )
4 ax-13 1686 . . . 4  |-  ( z  =  w  ->  (
z  e.  y  ->  w  e.  y )
)
53, 4imim12d 68 . . 3  |-  ( z  =  w  ->  (
( z  e.  x  ->  z  e.  y )  ->  ( w  e.  x  ->  w  e.  y ) ) )
65eximi 1563 . 2  |-  ( E. z  z  =  w  ->  E. z ( ( z  e.  x  -> 
z  e.  y )  ->  ( w  e.  x  ->  w  e.  y ) ) )
71, 6ax-mp 8 1  |-  E. z
( ( z  e.  x  ->  z  e.  y )  ->  (
w  e.  x  ->  w  e.  y )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1528
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529
  Copyright terms: Public domain W3C validator