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Theorem ax6o 1766
Description: Show that the original axiom ax-6o 2213 can be derived from ax-6 1744 and others. See ax6 2223 for the rederivation of ax-6 1744 from ax-6o 2213.

Normally, ax6o 1766 should be used rather than ax-6o 2213, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.)

Assertion
Ref Expression
ax6o  |-  ( -. 
A. x  -.  A. x ph  ->  ph )

Proof of Theorem ax6o
StepHypRef Expression
1 sp 1763 . 2  |-  ( A. x ph  ->  ph )
2 ax-6 1744 . 2  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
31, 2nsyl4 136 1  |-  ( -. 
A. x  -.  A. x ph  ->  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1549
This theorem is referenced by:  a6e  1767  modal-b  1768  hbntOLD  1800  nfndOLD  1810  equsalhwOLD  1861  ax9o  1954  hbntg  25425  ax4567  27559  hbntal  28567  ax9oNEW7  29396
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-ex 1551
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