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Theorem axc5c711toc7 2228
Description: Re-derivation of ax-c7 2194 from axc5c711 2226. Note that ax-c7 2194 and ax-11 1782 are not used by the re-derivation. The use of alimi 1605 (which uses ax-c5 2192) is allowed since we have already proved axc5c711toc5 2227. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc5c711toc7  |-  ( -. 
A. x  -.  A. x ph  ->  ph )

Proof of Theorem axc5c711toc7
StepHypRef Expression
1 hba1-o 2206 . . . . . 6  |-  ( A. x ph  ->  A. x A. x ph )
21con3i 135 . . . . 5  |-  ( -. 
A. x A. x ph  ->  -.  A. x ph )
32alimi 1605 . . . 4  |-  ( A. x  -.  A. x A. x ph  ->  A. x  -.  A. x ph )
43sps-o 2216 . . 3  |-  ( A. x A. x  -.  A. x A. x ph  ->  A. x  -.  A. x ph )
54con3i 135 . 2  |-  ( -. 
A. x  -.  A. x ph  ->  -.  A. x A. x  -.  A. x A. x ph )
6 pm2.21 108 . 2  |-  ( -. 
A. x A. x  -.  A. x A. x ph  ->  ( A. x A. x  -.  A. x A. x ph  ->  A. x ph ) )
7 axc5c711 2226 . 2  |-  ( ( A. x A. x  -.  A. x A. x ph  ->  A. x ph )  ->  ph )
85, 6, 73syl 20 1  |-  ( -. 
A. x  -.  A. x ph  ->  ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> 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-11 1782  ax-c5 2192  ax-c4 2193  ax-c7 2194
This theorem is referenced by:  axc5c711to11  2229
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