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Theorem ax9 1889
Description: Theorem showing that ax-9 1635 follows from the weaker version ax9v 1636. (Even though this theorem depends on ax-9 1635, all references of ax-9 1635 are made via ax9v 1636. An earlier version stated ax9v 1636 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-9 1635 so that all proofs can be traced back to ax9v 1636. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.)

Assertion
Ref Expression
ax9  |-  -.  A. x  -.  x  =  y

Proof of Theorem ax9
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 sp 1716 . . 3  |-  ( A. x  -.  x  =  y  ->  -.  x  =  y )
2 sp 1716 . . 3  |-  ( A. x  x  =  y  ->  x  =  y )
31, 2nsyl3 111 . 2  |-  ( A. x  x  =  y  ->  -.  A. x  -.  x  =  y )
4 ax9v 1636 . . 3  |-  -.  A. v  -.  v  =  y
5 dveeq2 1880 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  ( v  =  y  ->  A. x  v  =  y )
)
6 ax9v 1636 . . . . . . 7  |-  -.  A. x  -.  x  =  v
7 hba1 1719 . . . . . . . 8  |-  ( A. x  v  =  y  ->  A. x A. x  v  =  y )
8 sp 1716 . . . . . . . . . 10  |-  ( A. x  v  =  y  ->  v  =  y )
9 equequ2 1649 . . . . . . . . . 10  |-  ( v  =  y  ->  (
x  =  v  <->  x  =  y ) )
108, 9syl 15 . . . . . . . . 9  |-  ( A. x  v  =  y  ->  ( x  =  v  <-> 
x  =  y ) )
1110notbid 285 . . . . . . . 8  |-  ( A. x  v  =  y  ->  ( -.  x  =  v  <->  -.  x  =  y ) )
127, 11albidh 1577 . . . . . . 7  |-  ( A. x  v  =  y  ->  ( A. x  -.  x  =  v  <->  A. x  -.  x  =  y
) )
136, 12mtbii 293 . . . . . 6  |-  ( A. x  v  =  y  ->  -.  A. x  -.  x  =  y )
145, 13syl6com 31 . . . . 5  |-  ( v  =  y  ->  ( -.  A. x  x  =  y  ->  -.  A. x  -.  x  =  y
) )
1514con3i 127 . . . 4  |-  ( -.  ( -.  A. x  x  =  y  ->  -. 
A. x  -.  x  =  y )  ->  -.  v  =  y
)
1615alrimiv 1617 . . 3  |-  ( -.  ( -.  A. x  x  =  y  ->  -. 
A. x  -.  x  =  y )  ->  A. v  -.  v  =  y )
174, 16mt3 171 . 2  |-  ( -. 
A. x  x  =  y  ->  -.  A. x  -.  x  =  y
)
183, 17pm2.61i 156 1  |-  -.  A. x  -.  x  =  y
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   A.wal 1527
This theorem is referenced by:  ax9o  1890  a9e  1891  ax4567to4  27602  ax12-2  29103  ax12-4  29106
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