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Theorem ax467to6 2176
Description: Re-derivation of ax-6o 2142 from ax467 2174. Note that ax-6o 2142 and ax-7 1734 are not used by the re-derivation. The use of alimi 1559 (which uses ax-4 2140) is allowed since we have already proved ax467to4 2175. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax467to6  |-  ( -. 
A. x  -.  A. x ph  ->  ph )

Proof of Theorem ax467to6
StepHypRef Expression
1 hba1-o 2154 . . . . . 6  |-  ( A. x ph  ->  A. x A. x ph )
21con3i 127 . . . . 5  |-  ( -. 
A. x A. x ph  ->  -.  A. x ph )
32alimi 1559 . . . 4  |-  ( A. x  -.  A. x A. x ph  ->  A. x  -.  A. x ph )
43sps-o 2164 . . 3  |-  ( A. x A. x  -.  A. x A. x ph  ->  A. x  -.  A. x ph )
54con3i 127 . 2  |-  ( -. 
A. x  -.  A. x ph  ->  -.  A. x A. x  -.  A. x A. x ph )
6 pm2.21 100 . 2  |-  ( -. 
A. x A. x  -.  A. x A. x ph  ->  ( A. x A. x  -.  A. x A. x ph  ->  A. x ph ) )
7 ax467 2174 . 2  |-  ( ( A. x A. x  -.  A. x A. x ph  ->  A. x ph )  ->  ph )
85, 6, 73syl 18 1  |-  ( -. 
A. x  -.  A. x ph  ->  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1540
This theorem is referenced by:  ax467to7  2177
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-7 1734  ax-4 2140  ax-5o 2141  ax-6o 2142
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