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Theorem ax5o 1718
Description: Show that the original axiom ax-5o 2078 can be derived from ax-5 1545 and others. See ax5 2087 for the rederivation of ax-5 1545 from ax-5o 2078.

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 ax4 1717 . . . 4  |-  ( A. x  -.  A. x ph  ->  -.  A. x ph )
21con2i 114 . . 3  |-  ( A. x ph  ->  -.  A. x  -.  A. x ph )
3 ax-6 1704 . . 3  |-  ( -. 
A. x  -.  A. x ph  ->  A. x  -.  A. x  -.  A. x ph )
4 ax-6 1704 . . . . 5  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
54con1i 123 . . . 4  |-  ( -. 
A. x  -.  A. x ph  ->  A. x ph )
65alimi 1547 . . 3  |-  ( A. x  -.  A. x  -.  A. x ph  ->  A. x A. x ph )
72, 3, 63syl 20 . 2  |-  ( A. x ph  ->  A. x A. x ph )
8 ax-5 1545 . 2  |-  ( A. x ( A. x ph  ->  ps )  -> 
( A. x A. x ph  ->  A. x ps ) )
97, 8syl5 30 1  |-  ( A. x ( A. x ph  ->  ps )  -> 
( A. x ph  ->  A. x ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6   A.wal 1528
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-11 1716
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