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Theorem ax15 1961
Description: Axiom ax-15 2082 is redundant if we assume ax-17 1603. 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 1960 and ax-17 1603. 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 1704 . . . . 5  |-  ( -. 
A. z  z  =  y  ->  A. z  -.  A. z  z  =  y )
2 dveel2 1960 . . . . 5  |-  ( -. 
A. z  z  =  y  ->  ( w  e.  y  ->  A. z  w  e.  y )
)
31, 2hbim1 1732 . . . 4  |-  ( ( -.  A. z  z  =  y  ->  w  e.  y )  ->  A. z
( -.  A. z 
z  =  y  ->  w  e.  y )
)
4 elequ1 1687 . . . . 5  |-  ( w  =  x  ->  (
w  e.  y  <->  x  e.  y ) )
54imbi2d 307 . . . 4  |-  ( w  =  x  ->  (
( -.  A. z 
z  =  y  ->  w  e.  y )  <->  ( -.  A. z  z  =  y  ->  x  e.  y ) ) )
63, 5dvelim 1956 . . 3  |-  ( -. 
A. z  z  =  x  ->  ( ( -.  A. z  z  =  y  ->  x  e.  y )  ->  A. z
( -.  A. z 
z  =  y  ->  x  e.  y )
) )
7 nfa1 1756 . . . . 5  |-  F/ z A. z  z  =  y
87nfn 1765 . . . 4  |-  F/ z  -.  A. z  z  =  y
9819.21 1791 . . 3  |-  ( A. z ( -.  A. z  z  =  y  ->  x  e.  y )  <-> 
( -.  A. z 
z  =  y  ->  A. z  x  e.  y ) )
106, 9syl6ib 217 . 2  |-  ( -. 
A. z  z  =  x  ->  ( ( -.  A. z  z  =  y  ->  x  e.  y )  ->  ( -.  A. z  z  =  y  ->  A. z  x  e.  y )
) )
1110pm2.86d 93 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 1527
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532
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