Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ax13dfeq Unicode version

Theorem ax13dfeq 25179
Description: A version of ax-13 1719 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 1941 . 2  |-  E. z 
z  =  w
2 ax-13 1719 . . . 4  |-  ( w  =  z  ->  (
w  e.  x  -> 
z  e.  x ) )
32equcoms 1688 . . 3  |-  ( z  =  w  ->  (
w  e.  x  -> 
z  e.  x ) )
4 ax-13 1719 . . 3  |-  ( z  =  w  ->  (
z  e.  y  ->  w  e.  y )
)
53, 4imim12d 70 . 2  |-  ( z  =  w  ->  (
( z  e.  x  ->  z  e.  y )  ->  ( w  e.  x  ->  w  e.  y ) ) )
61, 5eximii 1584 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 1547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548
  Copyright terms: Public domain W3C validator