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Theorem ax5o 1693
Description: Show that the original axiom ax-5o 1694 can be derived from ax-5 1533 and others. See ax5 1695 for the rederivation of ax-5 1533 from ax-5o 1694.

Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114.

Normally, ax5o 1693 should be used rather than ax-5o 1694, except by theorems specifically studying the latter's properties. (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 ax-4 1692 . . . 4  |-  ( A. x  -.  A. x ph  ->  -.  A. x ph )
21con2i 114 . . 3  |-  ( A. x ph  ->  -.  A. x  -.  A. x ph )
3 ax-6 1534 . . 3  |-  ( -. 
A. x  -.  A. x ph  ->  A. x  -.  A. x  -.  A. x ph )
4 ax-6 1534 . . . . . 6  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
54con1i 123 . . . . 5  |-  ( -. 
A. x  -.  A. x ph  ->  A. x ph )
65ax-gen 1536 . . . 4  |-  A. x
( -.  A. x  -.  A. x ph  ->  A. x ph )
7 ax-5 1533 . . . 4  |-  ( A. x ( -.  A. x  -.  A. x ph  ->  A. x ph )  ->  ( A. x  -.  A. x  -.  A. x ph  ->  A. x A. x ph ) )
86, 7ax-mp 10 . . 3  |-  ( A. x  -.  A. x  -.  A. x ph  ->  A. x A. x ph )
92, 3, 83syl 20 . 2  |-  ( A. x ph  ->  A. x A. x ph )
10 ax-5 1533 . 2  |-  ( A. x ( A. x ph  ->  ps )  -> 
( A. x A. x ph  ->  A. x ps ) )
119, 10syl5 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 1532
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-4 1692
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