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Theorem ax12o 2010
Description: Derive set.mm's original ax-12o 2218 from the shorter ax-12 1950. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) (Proof shortened by Wolf Lammen, 29-Apr-2018.)
Assertion
Ref Expression
ax12o  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )

Proof of Theorem ax12o
StepHypRef Expression
1 nfeqf 2009 . . 3  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  x  =  y )
21nfrd 1779 . 2  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( x  =  y  ->  A. z  x  =  y )
)
32ex 424 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    /\ wa 359   A.wal 1549
This theorem is referenced by:  ax12  2019  ax12OLD  2020  dvelimvOLD  2028  hbae  2040  hbaeOLD  2041  nfeqfOLD  2050  dvelimfOLD  2065  dvelimhOLD  2068  dvelimALT  2209  ax11eq  2269  ax11indalem  2273  axi12  2414  axbnd  2415  axext4dist  25420  wl-aleq  26227
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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