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Theorem ax4567to5 27706
Description: Re-derivation of ax5o 1729 from ax4567 27704. Note that only propositional calculus is required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)
Assertion
Ref Expression
ax4567to5  |-  ( A. x ( A. x ph  ->  ps )  -> 
( A. x ph  ->  A. x ps )
)

Proof of Theorem ax4567to5
StepHypRef Expression
1 ax-1 5 . 2  |-  ( A. x ( A. x ph  ->  ps )  -> 
( ph  ->  A. x
( A. x ph  ->  ps ) ) )
2 ax-1 5 . 2  |-  ( (
ph  ->  A. x ( A. x ph  ->  ps )
)  ->  ( A. x A. x  -.  A. x A. x ( A. x ph  ->  ps )  ->  ( ph  ->  A. x
( A. x ph  ->  ps ) ) ) )
3 ax4567 27704 . 2  |-  ( ( A. x A. x  -.  A. x A. x
( A. x ph  ->  ps )  ->  ( ph  ->  A. x ( A. x ph  ->  ps )
) )  ->  ( A. x ph  ->  A. x ps ) )
41, 2, 33syl 18 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 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-7 1720  ax-11 1727
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