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Theorem ax10 31899
Description: Rederivation of axiom ax-10 1861 from ax-c7 31889 and other older axioms. See axc7 1885 for the derivation of ax-c7 31889 from ax-10 1861. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax10  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )

Proof of Theorem ax10
StepHypRef Expression
1 ax-c4 31888 . . 3  |-  ( A. x ( A. x  -.  A. x A. x ph  ->  -.  A. x ph )  ->  ( A. x  -.  A. x A. x ph  ->  A. x  -.  A. x ph )
)
2 ax-c5 31887 . . . 4  |-  ( A. x  -.  A. x A. x ph  ->  -.  A. x A. x ph )
3 ax-c4 31888 . . . . 5  |-  ( A. x ( A. x ph  ->  A. x ph )  ->  ( A. x ph  ->  A. x A. x ph ) )
4 id 22 . . . . 5  |-  ( A. x ph  ->  A. x ph )
53, 4mpg 1641 . . . 4  |-  ( A. x ph  ->  A. x A. x ph )
62, 5nsyl 121 . . 3  |-  ( A. x  -.  A. x A. x ph  ->  -.  A. x ph )
71, 6mpg 1641 . 2  |-  ( A. x  -.  A. x A. x ph  ->  A. x  -.  A. x ph )
8 ax-c7 31889 . 2  |-  ( -. 
A. x  -.  A. x A. x ph  ->  A. x ph )
97, 8nsyl4 142 1  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-c5 31887  ax-c4 31888  ax-c7 31889
This theorem is referenced by:  hba1-o  31901  axc5c711  31921  equidq  31927
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