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Theorem axc11-o 2269
Description: Show that ax-c11 2206 can be derived from ax-c11n 2207. An open problem is whether this theorem can be derived from ax-c11n 2207 and the others when ax-12 1798 is replaced with ax-c15 2208. See theorem axc11nfromc11 2244 for the rederivation of ax-c11n 2207 from axc11 2022.

Normally, axc11 2022 should be used rather than ax-c11 2206 or axc11-o 2269, 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 2207 . 2  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
2 ax12 2222 . . . 4  |-  ( y  =  x  ->  ( A. x ph  ->  A. y
( y  =  x  ->  ph ) ) )
32equcoms 1739 . . 3  |-  ( x  =  y  ->  ( A. x ph  ->  A. y
( y  =  x  ->  ph ) ) )
43sps-o 2226 . 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 1611 . 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 1372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-11 1786  ax-c5 2202  ax-c4 2203  ax-c7 2204  ax-c11 2206  ax-c11n 2207  ax-c15 2208  ax-c9 2209
This theorem depends on definitions:  df-bi 185  df-ex 1592
This theorem is referenced by: (None)
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