MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax12o Unicode version

Theorem ax12o 1877
Description: Derive set.mm's original ax-12o 2083 from the shorter ax-12 1868. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.)
Assertion
Ref Expression
ax12o  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )

Proof of Theorem ax12o
Dummy variables  w  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax12v 1869 . . 3  |-  ( -.  z  =  y  -> 
( y  =  w  ->  A. z  y  =  w ) )
2 ax12v 1869 . . 3  |-  ( -.  z  =  y  -> 
( y  =  v  ->  A. z  y  =  v ) )
31, 2ax12olem4 1873 . 2  |-  ( -.  z  =  y  -> 
( -.  A. z  -.  y  =  w  ->  A. z  y  =  w ) )
4 ax12v 1869 . . 3  |-  ( -.  z  =  x  -> 
( x  =  w  ->  A. z  x  =  w ) )
5 ax12v 1869 . . 3  |-  ( -.  z  =  x  -> 
( x  =  v  ->  A. z  x  =  v ) )
64, 5ax12olem4 1873 . 2  |-  ( -.  z  =  x  -> 
( -.  A. z  -.  x  =  w  ->  A. z  x  =  w ) )
73, 6ax12olem7 1876 1  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1529
This theorem is referenced by:  dvelimv  1881  hbae  1895  nfeqf  1900  dvelimh  1906  dvelimf  1939  dvelimALT  2074  ax11eq  2134  ax11indalem  2138  axext4dist  24159  ax12-2  29176  ax12-4  29179  ax10lem17ALT  29196  a12stdy4  29202  a12lem1  29203  ax9lem17  29229
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1531
  Copyright terms: Public domain W3C validator