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Theorem axc9 2105
Description: Derive set.mm's original ax-c9 32431 from the shorter ax-13 2057. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) (Proof shortened by Wolf Lammen, 29-Apr-2018.)
Assertion
Ref Expression
axc9  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )

Proof of Theorem axc9
StepHypRef Expression
1 nfeqf 2104 . . 3  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  x  =  y )
21nfrd 1930 . 2  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( x  =  y  ->  A. z  x  =  y )
)
32ex 435 1  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370   A.wal 1435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2057
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662
This theorem is referenced by:  ax13  2106  hbae  2114  axi12  2399  axbnd  2400  axext4dist  30454  bj-hbaeb2  31390  wl-aleq  31832  ax12eq  32481  ax12indalem  32485
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