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Theorem ax4567to4 27705
Description: Re-derivation of sp 1728 from ax4567 27704. Note that ax9 1902 is used for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)
Assertion
Ref Expression
ax4567to4  |-  ( A. x ph  ->  ph )

Proof of Theorem ax4567to4
StepHypRef Expression
1 ax9 1902 . . 3  |-  -.  A. x  -.  x  =  y
2 pm2.21 100 . . . 4  |-  ( -. 
ph  ->  ( ph  ->  A. x ( A. x ph  ->  -.  x  =  y ) ) )
3 ax-1 5 . . . 4  |-  ( (
ph  ->  A. x ( A. x ph  ->  -.  x  =  y ) )  ->  ( A. x A. x  -.  A. x A. x ( A. x ph  ->  -.  x  =  y )  ->  ( ph  ->  A. x ( A. x ph  ->  -.  x  =  y ) ) ) )
4 ax4567 27704 . . . 4  |-  ( ( A. x A. x  -.  A. x A. x
( A. x ph  ->  -.  x  =  y )  ->  ( ph  ->  A. x ( A. x ph  ->  -.  x  =  y ) ) )  ->  ( A. x ph  ->  A. x  -.  x  =  y
) )
52, 3, 43syl 18 . . 3  |-  ( -. 
ph  ->  ( A. x ph  ->  A. x  -.  x  =  y ) )
61, 5mtoi 169 . 2  |-  ( -. 
ph  ->  -.  A. x ph )
76con4i 122 1  |-  ( A. x ph  ->  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530    = wceq 1632
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  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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