MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax9 Unicode version

Theorem ax9 1893
Description: Theorem showing that ax-9 1638 follows from the weaker version ax9v 1639. (Even though this theorem depends on ax-9 1638, all references of ax-9 1638 are made via ax9v 1639. An earlier version stated ax9v 1639 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-9 1638 so that all proofs can be traced back to ax9v 1639. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.)

Assertion
Ref Expression
ax9  |-  -.  A. x  -.  x  =  y
Dummy variable  v is distinct from all other variables.

Proof of Theorem ax9
StepHypRef Expression
1 sp 1719 . . 3  |-  ( A. x  -.  x  =  y  ->  -.  x  =  y )
2 sp 1719 . . 3  |-  ( A. x  x  =  y  ->  x  =  y )
31, 2nsyl3 113 . 2  |-  ( A. x  x  =  y  ->  -.  A. x  -.  x  =  y )
4 ax9v 1639 . . 3  |-  -.  A. v  -.  v  =  y
5 dveeq2 1884 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  ( v  =  y  ->  A. x  v  =  y )
)
6 ax9v 1639 . . . . . . 7  |-  -.  A. x  -.  x  =  v
7 hba1 1723 . . . . . . . 8  |-  ( A. x  v  =  y  ->  A. x A. x  v  =  y )
8 sp 1719 . . . . . . . . . 10  |-  ( A. x  v  =  y  ->  v  =  y )
9 equequ2 1652 . . . . . . . . . 10  |-  ( v  =  y  ->  (
x  =  v  <->  x  =  y ) )
108, 9syl 17 . . . . . . . . 9  |-  ( A. x  v  =  y  ->  ( x  =  v  <-> 
x  =  y ) )
1110notbid 287 . . . . . . . 8  |-  ( A. x  v  =  y  ->  ( -.  x  =  v  <->  -.  x  =  y ) )
127, 11albidh 1579 . . . . . . 7  |-  ( A. x  v  =  y  ->  ( A. x  -.  x  =  v  <->  A. x  -.  x  =  y
) )
136, 12mtbii 295 . . . . . 6  |-  ( A. x  v  =  y  ->  -.  A. x  -.  x  =  y )
145, 13syl6com 33 . . . . 5  |-  ( v  =  y  ->  ( -.  A. x  x  =  y  ->  -.  A. x  -.  x  =  y
) )
1514con3i 129 . . . 4  |-  ( -.  ( -.  A. x  x  =  y  ->  -. 
A. x  -.  x  =  y )  ->  -.  v  =  y
)
1615alrimiv 1619 . . 3  |-  ( -.  ( -.  A. x  x  =  y  ->  -. 
A. x  -.  x  =  y )  ->  A. v  -.  v  =  y )
174, 16mt3 173 . 2  |-  ( -. 
A. x  x  =  y  ->  -.  A. x  -.  x  =  y
)
183, 17pm2.61i 158 1  |-  -.  A. x  -.  x  =  y
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178   A.wal 1529
This theorem is referenced by:  ax9o  1894  a9e  1895  ax4567to4  27003  ax12-2  28372  ax12-4  28375
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1531
  Copyright terms: Public domain W3C validator