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Theorem ax10o-o 2230
Description: Show that ax-10o 2166 can be derived from ax-10 2167. An open problem is whether this theorem can be derived from ax-10 2167 and the others when ax-11 1753 is replaced with ax-11o 2168. See theorem ax10from10o 2204 for the rederivation of ax-10 2167 from ax10o 1983.

Normally, ax10o 1983 should be used rather than ax-10o 2166 or ax10o-o 2230, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof of Theorem ax10o-o
StepHypRef Expression
1 ax-10 2167 . 2  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
2 ax11 2182 . . . 4  |-  ( y  =  x  ->  ( A. x ph  ->  A. y
( y  =  x  ->  ph ) ) )
32equcoms 1688 . . 3  |-  ( x  =  y  ->  ( A. x ph  ->  A. y
( y  =  x  ->  ph ) ) )
43sps-o 2186 . 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 1567 . 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 4   A.wal 1546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-7 1741  ax-4 2162  ax-5o 2163  ax-6o 2164  ax-10o 2166  ax-10 2167  ax-11o 2168  ax-12o 2169
This theorem depends on definitions:  df-bi 178  df-ex 1548
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