MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax6 Unicode version

Theorem ax6 2088
Description: Rederivation of axiom ax-6 1705 from ax-6o 2078 and other older axioms. See ax6o 1725 for the derivation of ax-6o 2078 from ax-6 1705. (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 2077 . . 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 2076 . . . 4  |-  ( A. x  -.  A. x A. x ph  ->  -.  A. x A. x ph )
3 ax-5o 2077 . . . . 5  |-  ( A. x ( A. x ph  ->  A. x ph )  ->  ( A. x ph  ->  A. x A. x ph ) )
4 id 19 . . . . 5  |-  ( A. x ph  ->  A. x ph )
53, 4mpg 1537 . . . 4  |-  ( A. x ph  ->  A. x A. x ph )
62, 5nsyl 113 . . 3  |-  ( A. x  -.  A. x A. x ph  ->  -.  A. x ph )
71, 6mpg 1537 . 2  |-  ( A. x  -.  A. x A. x ph  ->  A. x  -.  A. x ph )
8 ax-6o 2078 . 2  |-  ( -. 
A. x  -.  A. x A. x ph  ->  A. x ph )
97, 8nsyl4 134 1  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1529
This theorem is referenced by:  hba1-o  2090  ax467  2110  equidq  2116
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-4 2076  ax-5o 2077  ax-6o 2078
  Copyright terms: Public domain W3C validator