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Theorem ax10o 1835
Description: Show that ax-10o 1836 can be derived from ax-10 1678. An open problem is whether this theorem can be derived from ax-10 1678 and the others when ax-11 1624 is replaced with ax-11o 1941. See theorem ax10from10o 1837 for the rederivation of ax-10 1678 from ax10o 1835.

Normally, ax10o 1835 should be used rather than ax-10o 1836, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.)

Assertion
Ref Expression
ax10o  |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ph )
)

Proof of Theorem ax10o
StepHypRef Expression
1 ax-10 1678 . 2  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
2 ax-11 1624 . . . 4  |-  ( y  =  x  ->  ( A. x ph  ->  A. y
( y  =  x  ->  ph ) ) )
32equcoms 1825 . . 3  |-  ( x  =  y  ->  ( A. x ph  ->  A. y
( y  =  x  ->  ph ) ) )
43a4s 1700 . 2  |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ( y  =  x  ->  ph )
) )
5 pm2.27 37 . . 3  |-  ( y  =  x  ->  (
( y  =  x  ->  ph )  ->  ph )
)
65al2imi 1549 . 2  |-  ( A. y  y  =  x  ->  ( A. y ( y  =  x  ->  ph )  ->  A. y ph ) )
71, 4, 6sylsyld 54 1  |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 6   A.wal 1532
This theorem is referenced by:  hbae  1841  dvelimfALT  1854  dral1  1856  nd1  8163  nd2  8164  axpowndlem3  8175  a9e2eq  27360  a9e2eqVD  27717  2sb5ndVD  27720
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-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-10 1678  ax-9 1684  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-nf 1540
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