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

Theorem ax12o 1875
Description: Derive set.mm's original ax-12o 2081 from the shorter ax-12 1866. (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 1867 . . 3  |-  ( -.  z  =  y  -> 
( y  =  w  ->  A. z  y  =  w ) )
2 ax12v 1867 . . 3  |-  ( -.  z  =  y  -> 
( y  =  v  ->  A. z  y  =  v ) )
31, 2ax12olem4 1871 . 2  |-  ( -.  z  =  y  -> 
( -.  A. z  -.  y  =  w  ->  A. z  y  =  w ) )
4 ax12v 1867 . . 3  |-  ( -.  z  =  x  -> 
( x  =  w  ->  A. z  x  =  w ) )
5 ax12v 1867 . . 3  |-  ( -.  z  =  x  -> 
( x  =  v  ->  A. z  x  =  v ) )
64, 5ax12olem4 1871 . 2  |-  ( -.  z  =  x  -> 
( -.  A. z  -.  x  =  w  ->  A. z  x  =  w ) )
73, 6ax12olem7 1874 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 1527
This theorem is referenced by:  dvelimv  1879  hbae  1893  nfeqf  1898  dvelimh  1904  dvelimf  1937  dvelimALT  2072  ax11eq  2132  ax11indalem  2136  axext4dist  24157  ax12-2  29103  ax12-4  29106  ax10lem17ALT  29123  a12stdy4  29129  a12lem1  29130  ax9lem17  29156
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529
  Copyright terms: Public domain W3C validator