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Theorem ax15 2054
Description: Axiom ax-15 2177 is redundant if we assume ax-17 1623. Remark 9.6 in [Megill] p. 448 (p. 16 of the preprint), regarding axiom scheme C14'.

Note that  w is a dummy variable introduced in the proof. On the web page, it is implicitly assumed to be distinct from all other variables. (This is made explicit in the database file set.mm). Its purpose is to satisfy the distinct variable requirements of dveel2 2053 and ax-17 1623. By the end of the proof it has vanished, and the final theorem has no distinct variable requirements. (Contributed by NM, 29-Jun-1995.) (Proof modification is discouraged.)

Assertion
Ref Expression
ax15  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  e.  y  ->  A. z  x  e.  y )
) )

Proof of Theorem ax15
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 hbn1 1737 . . . . 5  |-  ( -. 
A. z  z  =  y  ->  A. z  -.  A. z  z  =  y )
2 dveel2 2053 . . . . 5  |-  ( -. 
A. z  z  =  y  ->  ( w  e.  y  ->  A. z  w  e.  y )
)
31, 2hbim1 1819 . . . 4  |-  ( ( -.  A. z  z  =  y  ->  w  e.  y )  ->  A. z
( -.  A. z 
z  =  y  ->  w  e.  y )
)
4 elequ1 1720 . . . . 5  |-  ( w  =  x  ->  (
w  e.  y  <->  x  e.  y ) )
54imbi2d 308 . . . 4  |-  ( w  =  x  ->  (
( -.  A. z 
z  =  y  ->  w  e.  y )  <->  ( -.  A. z  z  =  y  ->  x  e.  y ) ) )
63, 5dvelim 2050 . . 3  |-  ( -. 
A. z  z  =  x  ->  ( ( -.  A. z  z  =  y  ->  x  e.  y )  ->  A. z
( -.  A. z 
z  =  y  ->  x  e.  y )
) )
7 nfa1 1796 . . . . 5  |-  F/ z A. z  z  =  y
87nfn 1801 . . . 4  |-  F/ z  -.  A. z  z  =  y
9819.21 1804 . . 3  |-  ( A. z ( -.  A. z  z  =  y  ->  x  e.  y )  <-> 
( -.  A. z 
z  =  y  ->  A. z  x  e.  y ) )
106, 9syl6ib 218 . 2  |-  ( -. 
A. z  z  =  x  ->  ( ( -.  A. z  z  =  y  ->  x  e.  y )  ->  ( -.  A. z  z  =  y  ->  A. z  x  e.  y )
) )
1110pm2.86d 95 1  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  e.  y  ->  A. z  x  e.  y )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1546
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-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551
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