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Theorem ax10 2025
Description: Derive set.mm's original ax-10 2217 from others. (Contributed by NM, 25-Jul-2015.) (Revised by NM, 7-Nov-2015.) (Proof shortened by Wolf Lammen, 6-Mar-2018.)
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 a9ev 1668 . . 3  |-  E. z 
z  =  x
2 equcomi 1691 . . . . . 6  |-  ( z  =  x  ->  x  =  z )
3 dveeq1 2021 . . . . . 6  |-  ( -. 
A. y  y  =  x  ->  ( x  =  z  ->  A. y  x  =  z )
)
42, 3syl5com 28 . . . . 5  |-  ( z  =  x  ->  ( -.  A. y  y  =  x  ->  A. y  x  =  z )
)
5 ax10o2 2024 . . . . . 6  |-  ( A. x  x  =  y  ->  ( A. y  x  =  z  ->  A. x  x  =  z )
)
6 ax10lem2 2023 . . . . . 6  |-  ( A. x  x  =  z  ->  A. y  y  =  x )
75, 6syl6 31 . . . . 5  |-  ( A. x  x  =  y  ->  ( A. y  x  =  z  ->  A. y 
y  =  x ) )
84, 7syl9 68 . . . 4  |-  ( z  =  x  ->  ( A. x  x  =  y  ->  ( -.  A. y  y  =  x  ->  A. y  y  =  x ) ) )
98exlimiv 1644 . . 3  |-  ( E. z  z  =  x  ->  ( A. x  x  =  y  ->  ( -.  A. y  y  =  x  ->  A. y 
y  =  x ) ) )
101, 9ax-mp 8 . 2  |-  ( A. x  x  =  y  ->  ( -.  A. y 
y  =  x  ->  A. y  y  =  x ) )
1110pm2.18d 105 1  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1549   E.wex 1550
This theorem is referenced by:  aecom  2035  aecoms  2036  naecoms  2037  ax10oOLD  2039  sbcom  2164  axi10  2414  2sb5ndVD  29022  e2ebindVD  29024  e2ebindALT  29041  2sb5ndALT  29044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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