MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax67to6 Unicode version

Theorem ax67to6 2201
Description: Re-derivation of ax-6o 2171 from ax67 2199. Note that ax-6o 2171 and ax-7 1741 are not used by the re-derivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax67to6  |-  ( -. 
A. x  -.  A. x ph  ->  ph )

Proof of Theorem ax67to6
StepHypRef Expression
1 hba1-o 2183 . . . . 5  |-  ( A. x ph  ->  A. x A. x ph )
21con3i 129 . . . 4  |-  ( -. 
A. x A. x ph  ->  -.  A. x ph )
32alimi 1565 . . 3  |-  ( A. x  -.  A. x A. x ph  ->  A. x  -.  A. x ph )
43con3i 129 . 2  |-  ( -. 
A. x  -.  A. x ph  ->  -.  A. x  -.  A. x A. x ph )
5 ax67 2199 . 2  |-  ( -. 
A. x  -.  A. x A. x ph  ->  A. x ph )
6 ax-4 2169 . 2  |-  ( A. x ph  ->  ph )
74, 5, 63syl 19 1  |-  ( -. 
A. x  -.  A. x ph  ->  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1546
This theorem is referenced by:  ax67to7  2202
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-7 1741  ax-4 2169  ax-5o 2170  ax-6o 2171
  Copyright terms: Public domain W3C validator