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Theorem ax6o 1758
Description: Show that the original axiom ax-6o 2164 can be derived from ax-6 1736 and others. See ax6 2174 for the rederivation of ax-6 1736 from ax-6o 2164.

Normally, ax6o 1758 should be used rather than ax-6o 2164, 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 1755 . 2  |-  ( A. x ph  ->  ph )
2 ax-6 1736 . 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 1546
This theorem is referenced by:  a6e  1759  modal-b  1760  hbnt  1783  nfndOLD  1800  equsalhwOLD  1851  ax9o  1943  hbntg  25179  ax4567  27263  hbntal  27976  ax9oNEW7  28800
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-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-11 1753
This theorem depends on definitions:  df-bi 178  df-ex 1548
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