MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax467to6 Unicode version

Theorem ax467to6 2110
Description: Re-derivation of ax-6o 2076 from ax467 2108. Note that ax-6o 2076 and ax-7 1708 are not used by the re-derivation. The use of alimi 1546 (which uses ax-4 2074) is allowed since we have already proved ax467to4 2109. (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 2088 . . . . . 6  |-  ( A. x ph  ->  A. x A. x ph )
21con3i 127 . . . . 5  |-  ( -. 
A. x A. x ph  ->  -.  A. x ph )
32alimi 1546 . . . 4  |-  ( A. x  -.  A. x A. x ph  ->  A. x  -.  A. x ph )
43sps-o 2098 . . 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 2108 . 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 1527
This theorem is referenced by:  ax467to7  2111
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-7 1708  ax-4 2074  ax-5o 2075  ax-6o 2076
  Copyright terms: Public domain W3C validator