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

Theorem ax9 1683
Description: Theorem showing that ax-9 1684 follows from the weaker version ax-9v 1632.

This theorem normally should not be referenced in any later proof. Instead, the use of ax-9 1684 below is preferred, since it is easier to work with (it has no distinct variable conditions) and it is the standard version we have adopted. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.) (New usage is discouraged.)

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

Proof of Theorem ax9
StepHypRef Expression
1 ax12o10lem3 1637 . . 3  |-  ( A. x  -.  x  =  y  ->  -.  x  =  y )
2 ax12o10lem3 1637 . . 3  |-  ( A. x  x  =  y  ->  x  =  y )
31, 2nsyl3 113 . 2  |-  ( A. x  x  =  y  ->  -.  A. x  -.  x  =  y )
4 ax-9v 1632 . . 3  |-  -.  A. v  -.  v  =  y
5 ax-17 1628 . . . 4  |-  ( -.  ( -.  A. x  x  =  y  ->  -. 
A. x  -.  x  =  y )  ->  A. v  -.  ( -.  A. x  x  =  y  ->  -.  A. x  -.  x  =  y
) )
6 ax10lem25 1674 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  ( v  =  y  ->  A. x  v  =  y )
)
7 ax-9v 1632 . . . . . . 7  |-  -.  A. x  -.  x  =  v
8 ax12o10lem5 1639 . . . . . . . 8  |-  ( A. x  v  =  y  ->  A. x A. x  v  =  y )
9 ax12o10lem3 1637 . . . . . . . . . 10  |-  ( A. x  v  =  y  ->  v  =  y )
10 ax10lem16 1665 . . . . . . . . . 10  |-  ( v  =  y  ->  (
x  =  v  <->  x  =  y ) )
119, 10syl 17 . . . . . . . . 9  |-  ( A. x  v  =  y  ->  ( x  =  v  <-> 
x  =  y ) )
1211notbid 287 . . . . . . . 8  |-  ( A. x  v  =  y  ->  ( -.  x  =  v  <->  -.  x  =  y ) )
138, 12albidh 1589 . . . . . . 7  |-  ( A. x  v  =  y  ->  ( A. x  -.  x  =  v  <->  A. x  -.  x  =  y
) )
147, 13mtbii 295 . . . . . 6  |-  ( A. x  v  =  y  ->  -.  A. x  -.  x  =  y )
156, 14syl6com 33 . . . . 5  |-  ( v  =  y  ->  ( -.  A. x  x  =  y  ->  -.  A. x  -.  x  =  y
) )
1615con3i 129 . . . 4  |-  ( -.  ( -.  A. x  x  =  y  ->  -. 
A. x  -.  x  =  y )  ->  -.  v  =  y
)
175, 16alrimih 1553 . . 3  |-  ( -.  ( -.  A. x  x  =  y  ->  -. 
A. x  -.  x  =  y )  ->  A. v  -.  v  =  y )
184, 17mt3 173 . 2  |-  ( -. 
A. x  x  =  y  ->  -.  A. x  -.  x  =  y
)
193, 18pm2.61i 158 1  |-  -.  A. x  -.  x  =  y
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178   A.wal 1532
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
  Copyright terms: Public domain W3C validator