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Theorem ax10 1885
Description: Derive set.mm's original ax-10 2082 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 )
Dummy variable  z is distinct from all other variables.

Proof of Theorem ax10
StepHypRef Expression
1 ax9v 1638 . 2  |-  -.  A. z  -.  z  =  x
2 df-ex 1530 . . 3  |-  ( E. z  z  =  x  <->  -.  A. z  -.  z  =  x )
3 dveeq2 1881 . . . . . . . 8  |-  ( -. 
A. y  y  =  x  ->  ( z  =  x  ->  A. y 
z  =  x ) )
43imp 420 . . . . . . 7  |-  ( ( -.  A. y  y  =  x  /\  z  =  x )  ->  A. y 
z  =  x )
5 ax10lem6 1884 . . . . . . . 8  |-  ( A. x  x  =  y  ->  ( A. y  z  =  x  ->  A. x  z  =  x )
)
6 equcomi 1647 . . . . . . . . 9  |-  ( z  =  x  ->  x  =  z )
76alimi 1547 . . . . . . . 8  |-  ( A. x  z  =  x  ->  A. x  x  =  z )
85, 7syl6 31 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( A. y  z  =  x  ->  A. x  x  =  z )
)
9 ax10lem5 1883 . . . . . . 7  |-  ( A. x  x  =  z  ->  A. y  y  =  x )
104, 8, 9syl56 32 . . . . . 6  |-  ( A. x  x  =  y  ->  ( ( -.  A. y  y  =  x  /\  z  =  x
)  ->  A. y 
y  =  x ) )
1110exp3acom23 1364 . . . . 5  |-  ( A. x  x  =  y  ->  ( z  =  x  ->  ( -.  A. y  y  =  x  ->  A. y  y  =  x ) ) )
12 pm2.18 104 . . . . 5  |-  ( ( -.  A. y  y  =  x  ->  A. y 
y  =  x )  ->  A. y  y  =  x )
1311, 12syl6 31 . . . 4  |-  ( A. x  x  =  y  ->  ( z  =  x  ->  A. y  y  =  x ) )
1413exlimdv 1665 . . 3  |-  ( A. x  x  =  y  ->  ( E. z  z  =  x  ->  A. y 
y  =  x ) )
152, 14syl5bir 211 . 2  |-  ( A. x  x  =  y  ->  ( -.  A. z  -.  z  =  x  ->  A. y  y  =  x ) )
161, 15mpi 18 1  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360   A.wal 1528   E.wex 1529
This theorem is referenced by:  alequcom  1887  ax10o  1893  e2ebindALT  27974
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1530
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