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Theorem ax6 2197
Description: Rederivation of axiom ax-6 1740 from ax-6o 2187 and other older axioms. See ax6o 1762 for the derivation of ax-6o 2187 from ax-6 1740. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax6  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )

Proof of Theorem ax6
StepHypRef Expression
1 ax-5o 2186 . . 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-4 2185 . . . 4  |-  ( A. x  -.  A. x A. x ph  ->  -.  A. x A. x ph )
3 ax-5o 2186 . . . . 5  |-  ( A. x ( A. x ph  ->  A. x ph )  ->  ( A. x ph  ->  A. x A. x ph ) )
4 id 20 . . . . 5  |-  ( A. x ph  ->  A. x ph )
53, 4mpg 1554 . . . 4  |-  ( A. x ph  ->  A. x A. x ph )
62, 5nsyl 115 . . 3  |-  ( A. x  -.  A. x A. x ph  ->  -.  A. x ph )
71, 6mpg 1554 . 2  |-  ( A. x  -.  A. x A. x ph  ->  A. x  -.  A. x ph )
8 ax-6o 2187 . 2  |-  ( -. 
A. x  -.  A. x A. x ph  ->  A. x ph )
97, 8nsyl4 136 1  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1546
This theorem is referenced by:  hba1-o  2199  ax467  2219  equidq  2225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-4 2185  ax-5o 2186  ax-6o 2187
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