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Theorem ax10o 1650
Description: Show that ax-10o 1651 can be derived from ax-10 1441. An open problem is whether this theorem can be derived from ax-10 1441 and the others when ax-11 1442 is replaced with ax-11o 1751. See theorem ax10 1652 for the rederivation of ax-10 1441 from ax10o 1650.

Normally, ax10o 1650 should be used rather than ax-10o 1651, 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 1441 . 2  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
2 ax-11 1442 . . . 4  |-  ( y  =  x  ->  ( A. x ph  ->  A. y
( y  =  x  ->  ph ) ) )
32equcoms 1641 . . 3  |-  ( x  =  y  ->  ( A. x ph  ->  A. y
( y  =  x  ->  ph ) ) )
43sps 1475 . 2  |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ( y  =  x  ->  ph )
) )
5 pm2.27 39 . . 3  |-  ( y  =  x  ->  (
( y  =  x  ->  ph )  ->  ph )
)
65al2imi 1392 . 2  |-  ( A. y  y  =  x  ->  ( A. y ( y  =  x  ->  ph )  ->  A. y ph ) )
71, 4, 6sylsyld 57 1  |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-5 1381  ax-gen 1383  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  hbae  1653  dral1  1665
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