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Theorem ax10 1884
Description: Derive set.mm's original ax-10 2079 from others. (Contributed by NM, 25-Jul-2015.) (Revised by NM, 7-Nov-2015.)
Assertion
Ref Expression
ax10  |-  ( A. x  x  =  y  ->  A. y  y  =  x )

Proof of Theorem ax10
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ax9v 1636 . 2  |-  -.  A. z  -.  z  =  x
2 df-ex 1529 . . 3  |-  ( E. z  z  =  x  <->  -.  A. z  -.  z  =  x )
3 dveeq2 1880 . . . . . . . 8  |-  ( -. 
A. y  y  =  x  ->  ( z  =  x  ->  A. y 
z  =  x ) )
43imp 418 . . . . . . 7  |-  ( ( -.  A. y  y  =  x  /\  z  =  x )  ->  A. y 
z  =  x )
5 ax10lem6 1883 . . . . . . . 8  |-  ( A. x  x  =  y  ->  ( A. y  z  =  x  ->  A. x  z  =  x )
)
6 equcomi 1646 . . . . . . . . 9  |-  ( z  =  x  ->  x  =  z )
76alimi 1546 . . . . . . . 8  |-  ( A. x  z  =  x  ->  A. x  x  =  z )
85, 7syl6 29 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( A. y  z  =  x  ->  A. x  x  =  z )
)
9 ax10lem5 1882 . . . . . . 7  |-  ( A. x  x  =  z  ->  A. y  y  =  x )
104, 8, 9syl56 30 . . . . . 6  |-  ( A. x  x  =  y  ->  ( ( -.  A. y  y  =  x  /\  z  =  x
)  ->  A. y 
y  =  x ) )
1110exp3acom23 1362 . . . . 5  |-  ( A. x  x  =  y  ->  ( z  =  x  ->  ( -.  A. y  y  =  x  ->  A. y  y  =  x ) ) )
12 pm2.18 102 . . . . 5  |-  ( ( -.  A. y  y  =  x  ->  A. y 
y  =  x )  ->  A. y  y  =  x )
1311, 12syl6 29 . . . 4  |-  ( A. x  x  =  y  ->  ( z  =  x  ->  A. y  y  =  x ) )
1413exlimdv 1664 . . 3  |-  ( A. x  x  =  y  ->  ( E. z  z  =  x  ->  A. y 
y  =  x ) )
152, 14syl5bir 209 . 2  |-  ( A. x  x  =  y  ->  ( -.  A. z  -.  z  =  x  ->  A. y  y  =  x ) )
161, 15mpi 16 1  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1527   E.wex 1528
This theorem is referenced by:  aecom  1886  ax10o  1892  2sb5ndVD  28059  e2ebindVD  28061  e2ebindALT  28079  2sb5ndALT  28082
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
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