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Theorem ax10o 1708
Description: Show that ax-10o 1709 can be derived from ax-10 1498. An open problem is whether this theorem can be derived from ax-10 1498 and the others when ax-11 1499 is replaced with ax-11o 1816. See Theorem ax10 1710 for the rederivation of ax-10 1498 from ax10o 1708.

Normally, ax10o 1708 should be used rather than ax-10o 1709, 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 1498 . 2  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
2 ax-11 1499 . . . 4  |-  ( y  =  x  ->  ( A. x ph  ->  A. y
( y  =  x  ->  ph ) ) )
32equcoms 1701 . . 3  |-  ( x  =  y  ->  ( A. x ph  ->  A. y
( y  =  x  ->  ph ) ) )
43sps 1530 . 2  |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ( y  =  x  ->  ph )
) )
5 pm2.27 40 . . 3  |-  ( y  =  x  ->  (
( y  =  x  ->  ph )  ->  ph )
)
65al2imi 1451 . 2  |-  ( A. y  y  =  x  ->  ( A. y ( y  =  x  ->  ph )  ->  A. y ph ) )
71, 4, 6sylsyld 58 1  |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-5 1440  ax-gen 1442  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  hbae  1711  dral1  1723
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