MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axc14 Structured version   Unicode version

Theorem axc14 2164
Description: Axiom ax-c14 32216 is redundant if we assume ax-5 1748. 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 2163 and ax-5 1748. 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 1887 . . . . 5  |-  ( -. 
A. z  z  =  y  ->  A. z  -.  A. z  z  =  y )
2 dveel2 2163 . . . . 5  |-  ( -. 
A. z  z  =  y  ->  ( w  e.  y  ->  A. z  w  e.  y )
)
31, 2hbim1 1973 . . . 4  |-  ( ( -.  A. z  z  =  y  ->  w  e.  y )  ->  A. z
( -.  A. z 
z  =  y  ->  w  e.  y )
)
4 elequ1 1870 . . . . 5  |-  ( w  =  x  ->  (
w  e.  y  <->  x  e.  y ) )
54imbi2d 317 . . . 4  |-  ( w  =  x  ->  (
( -.  A. z 
z  =  y  ->  w  e.  y )  <->  ( -.  A. z  z  =  y  ->  x  e.  y ) ) )
63, 5dvelim 2132 . . 3  |-  ( -. 
A. z  z  =  x  ->  ( ( -.  A. z  z  =  y  ->  x  e.  y )  ->  A. z
( -.  A. z 
z  =  y  ->  x  e.  y )
) )
7 nfa1 1951 . . . . 5  |-  F/ z A. z  z  =  y
87nfn 1955 . . . 4  |-  F/ z  -.  A. z  z  =  y
9819.21 1959 . . 3  |-  ( A. z ( -.  A. z  z  =  y  ->  x  e.  y )  <-> 
( -.  A. z 
z  =  y  ->  A. z  x  e.  y ) )
106, 9syl6ib 229 . 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 1435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator