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Theorem ax10o-o 2142
Description: Show that ax-10o 2078 can be derived from ax-10 2079. An open problem is whether this theorem can be derived from ax-10 2079 and the others when ax-11 1715 is replaced with ax-11o 2080. See theorem ax10from10o 2116 for the rederivation of ax-10 2079 from ax10o 1892.

Normally, ax10o 1892 should be used rather than ax-10o 2078 or ax10o-o 2142, 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 2079 . 2  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
2 ax11 2094 . . . 4  |-  ( y  =  x  ->  ( A. x ph  ->  A. y
( y  =  x  ->  ph ) ) )
32equcoms 1651 . . 3  |-  ( x  =  y  ->  ( A. x ph  ->  A. y
( y  =  x  ->  ph ) ) )
43sps-o 2098 . 2  |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ( y  =  x  ->  ph )
) )
5 pm2.27 35 . . 3  |-  ( y  =  x  ->  (
( y  =  x  ->  ph )  ->  ph )
)
65al2imi 1548 . 2  |-  ( A. y  y  =  x  ->  ( A. y ( y  =  x  ->  ph )  ->  A. y ph ) )
71, 4, 6sylsyld 52 1  |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-7 1708  ax-4 2074  ax-5o 2075  ax-6o 2076  ax-10o 2078  ax-10 2079  ax-11o 2080  ax-12o 2081
This theorem depends on definitions:  df-bi 177
  Copyright terms: Public domain W3C validator