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Theorem axc14 2201
Description: Axiom ax-c14 32463 is redundant if we assume ax-5 1758. 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. Its purpose is to satisfy the distinct variable requirements of dveel2 2200 and ax-5 1758. 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
axc14  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  e.  y  ->  A. z  x  e.  y )
) )

Proof of Theorem axc14
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 hbn1 1916 . . . . 5  |-  ( -. 
A. z  z  =  y  ->  A. z  -.  A. z  z  =  y )
2 dveel2 2200 . . . . 5  |-  ( -. 
A. z  z  =  y  ->  ( w  e.  y  ->  A. z  w  e.  y )
)
31, 2hbim1 2001 . . . 4  |-  ( ( -.  A. z  z  =  y  ->  w  e.  y )  ->  A. z
( -.  A. z 
z  =  y  ->  w  e.  y )
)
4 elequ1 1894 . . . . 5  |-  ( w  =  x  ->  (
w  e.  y  <->  x  e.  y ) )
54imbi2d 318 . . . 4  |-  ( w  =  x  ->  (
( -.  A. z 
z  =  y  ->  w  e.  y )  <->  ( -.  A. z  z  =  y  ->  x  e.  y ) ) )
63, 5dvelim 2171 . . 3  |-  ( -. 
A. z  z  =  x  ->  ( ( -.  A. z  z  =  y  ->  x  e.  y )  ->  A. z
( -.  A. z 
z  =  y  ->  x  e.  y )
) )
7 nfa1 1979 . . . . 5  |-  F/ z A. z  z  =  y
87nfn 1983 . . . 4  |-  F/ z  -.  A. z  z  =  y
9819.21 1987 . . 3  |-  ( A. z ( -.  A. z  z  =  y  ->  x  e.  y )  <-> 
( -.  A. z 
z  =  y  ->  A. z  x  e.  y ) )
106, 9syl6ib 230 . 2  |-  ( -. 
A. z  z  =  x  ->  ( ( -.  A. z  z  =  y  ->  x  e.  y )  ->  ( -.  A. z  z  =  y  ->  A. z  x  e.  y )
) )
1110pm2.86d 102 1  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  e.  y  ->  A. z  x  e.  y )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668
This theorem is referenced by: (None)
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