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

Theorem ax12o 1964
Description: Derive set.mm's original ax-12o 2176 from the shorter ax-12 1939. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) (Revised by Wolf Lammen, 30-Jan-2018.)
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 1940 . . 3  |-  ( -.  z  =  x  -> 
( x  =  w  ->  A. z  x  =  w ) )
2 ax12v 1940 . . 3  |-  ( -.  z  =  x  -> 
( x  =  v  ->  A. z  x  =  v ) )
31, 2ax12olem3 1962 . 2  |-  ( -. 
A. z  z  =  x  ->  F/ z  x  =  w )
4 ax12v 1940 . . 3  |-  ( -.  z  =  y  -> 
( y  =  w  ->  A. z  y  =  w ) )
5 ax12v 1940 . . 3  |-  ( -.  z  =  y  -> 
( y  =  v  ->  A. z  y  =  v ) )
64, 5ax12olem3 1962 . 2  |-  ( -. 
A. z  z  =  y  ->  F/ z 
y  =  w )
73, 6ax12olem4 1963 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 1546
This theorem is referenced by:  ax12  1973  ax12OLD  1974  dvelimvOLD  1985  hbae  1997  nfeqf  2002  dvelimh  2003  dvelimf  2031  dvelimALT  2167  ax11eq  2227  ax11indalem  2231  axi12  2367  axext4dist  25181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551
  Copyright terms: Public domain W3C validator