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Theorem ax9o 1963
Description: Show that the original axiom ax-9o 2151 can be derived from ax9 1962 and others. See ax9from9o 2161 for the rederivation of ax9 1962 from ax-9o 2151.

Normally, ax9o 1963 should be used rather than ax-9o 2151, 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 ax9 1962 . . 3  |-  -.  A. x  -.  x  =  y
2 con3 126 . . . 4  |-  ( ( x  =  y  ->  A. x ph )  -> 
( -.  A. x ph  ->  -.  x  =  y ) )
32al2imi 1566 . . 3  |-  ( A. x ( x  =  y  ->  A. x ph )  ->  ( A. x  -.  A. x ph  ->  A. x  -.  x  =  y ) )
41, 3mtoi 169 . 2  |-  ( A. x ( x  =  y  ->  A. x ph )  ->  -.  A. x  -.  A. x ph )
5 ax6o 1756 . 2  |-  ( -. 
A. x  -.  A. x ph  ->  ph )
64, 5syl 15 1  |-  ( A. x ( x  =  y  ->  A. x ph )  ->  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1545
This theorem is referenced by:  equsal  1973  spimt  1987  cbv1h  1991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1547  df-nf 1550
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