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Theorem ax9o 1814
Description: Show that the original axiom ax-9o 1815 can be derived from ax-9 1684 and others. See ax9from9o 1816 for the rederivation of ax-9 1684 from ax-9o 1815.

Normally, ax9o 1814 should be used rather than ax-9o 1815, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.)

Assertion
Ref Expression
ax9o  |-  ( A. x ( x  =  y  ->  A. x ph )  ->  ph )

Proof of Theorem ax9o
StepHypRef Expression
1 ax-9 1684 . . 3  |-  -.  A. x  -.  x  =  y
2 con3 128 . . . 4  |-  ( ( x  =  y  ->  A. x ph )  -> 
( -.  A. x ph  ->  -.  x  =  y ) )
32al2imi 1549 . . 3  |-  ( A. x ( x  =  y  ->  A. x ph )  ->  ( A. x  -.  A. x ph  ->  A. x  -.  x  =  y ) )
41, 3mtoi 171 . 2  |-  ( A. x ( x  =  y  ->  A. x ph )  ->  -.  A. x  -.  A. x ph )
5 ax-4 1692 . . 3  |-  ( A. x ph  ->  ph )
6 ax-6 1534 . . 3  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
75, 6nsyl4 136 . 2  |-  ( -. 
A. x  -.  A. x ph  ->  ph )
84, 7syl 17 1  |-  ( A. x ( x  =  y  ->  A. x ph )  ->  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6   A.wal 1532
This theorem is referenced by:  equsal  1851  equsalh  1852  a4imt  1867  cbv1h  1871
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-gen 1536  ax-9 1684  ax-4 1692
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