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Theorem ax5o 1729
Description: Show that the original axiom ax-5o 2088 can be derived from ax-5 1547 and others. See ax5 2098 for the rederivation of ax-5 1547 from ax-5o 2088.

Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114. (Contributed by NM, 21-May-2008.) (Proof modification is discouraged.)

Assertion
Ref Expression
ax5o  |-  ( A. x ( A. x ph  ->  ps )  -> 
( A. x ph  ->  A. x ps )
)

Proof of Theorem ax5o
StepHypRef Expression
1 sp 1728 . . . 4  |-  ( A. x  -.  A. x ph  ->  -.  A. x ph )
21con2i 112 . . 3  |-  ( A. x ph  ->  -.  A. x  -.  A. x ph )
3 hbn1 1716 . . 3  |-  ( -. 
A. x  -.  A. x ph  ->  A. x  -.  A. x  -.  A. x ph )
4 hbn1 1716 . . . . 5  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
54con1i 121 . . . 4  |-  ( -. 
A. x  -.  A. x ph  ->  A. x ph )
65alimi 1549 . . 3  |-  ( A. x  -.  A. x  -.  A. x ph  ->  A. x A. x ph )
72, 3, 63syl 18 . 2  |-  ( A. x ph  ->  A. x A. x ph )
8 ax-5 1547 . 2  |-  ( A. x ( A. x ph  ->  ps )  -> 
( A. x A. x ph  ->  A. x ps ) )
97, 8syl5 28 1  |-  ( A. x ( A. x ph  ->  ps )  -> 
( A. x ph  ->  A. x ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530
This theorem is referenced by:  ax4567  27704
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727
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