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Theorem axc11-o 2259
Description: Show that ax-c11 2196 can be derived from ax-c11n 2197. An open problem is whether this theorem can be derived from ax-c11n 2197 and the others when ax-12 1794 is replaced with ax-c15 2198. See theorem axc11nfromc11 2234 for the rederivation of ax-c11n 2197 from axc11 2011.

Normally, axc11 2011 should be used rather than ax-c11 2196 or axc11-o 2259, 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
axc11-o  |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ph )
)

Proof of Theorem axc11-o
StepHypRef Expression
1 ax-c11n 2197 . 2  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
2 ax12 2212 . . . 4  |-  ( y  =  x  ->  ( A. x ph  ->  A. y
( y  =  x  ->  ph ) ) )
32equcoms 1735 . . 3  |-  ( x  =  y  ->  ( A. x ph  ->  A. y
( y  =  x  ->  ph ) ) )
43sps-o 2216 . 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 1607 . 2  |-  ( A. y  y  =  x  ->  ( A. y ( y  =  x  ->  ph )  ->  A. y ph ) )
71, 4, 6sylsyld 56 1  |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ph )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-11 1782  ax-c5 2192  ax-c4 2193  ax-c7 2194  ax-c11 2196  ax-c11n 2197  ax-c15 2198  ax-c9 2199
This theorem depends on definitions:  df-bi 185  df-ex 1588
This theorem is referenced by: (None)
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