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Theorem ax12o 1663
Description: Derive set.mm's original ax-12o 1664 from the shorter ax-12 1633.

Our current practice is to use axiom ax-12o 1664 from here on (except in the proofs of ax10 1677 and ax9 1683 below) instead of theorem ax12o 1663 in order to standardize the use ax-9 1684 instead of ax-9v 1632. Note that the derivation of ax-9 1684 from ax-9v 1632 (theorem ax9 1683 below) makes use of ax12o 1663; thus we use ax-9v 1632 to prove ax12o 1663 to avoid a circular argument . (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) (New usage is discouraged.)

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 ax12v 1634 . . 3  |-  ( -.  z  =  y  -> 
( y  =  w  ->  A. z  y  =  w ) )
2 ax12v 1634 . . 3  |-  ( -.  z  =  y  -> 
( y  =  v  ->  A. z  y  =  v ) )
31, 2ax12olem24 1658 . 2  |-  ( -.  z  =  y  -> 
( -.  A. z  -.  y  =  w  ->  A. z  y  =  w ) )
4 ax12v 1634 . . 3  |-  ( -.  z  =  x  -> 
( x  =  w  ->  A. z  x  =  w ) )
5 ax12v 1634 . . 3  |-  ( -.  z  =  x  -> 
( x  =  v  ->  A. z  x  =  v ) )
64, 5ax12olem24 1658 . 2  |-  ( -.  z  =  x  -> 
( -.  A. z  -.  x  =  w  ->  A. z  x  =  w ) )
73, 6ax12olem28 1662 1  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6   A.wal 1532
This theorem is referenced by:  ax10lem24  1673
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-9v 1632  ax-12 1633
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538
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