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Theorem ax4567to4 27593
Description: Re-derivation of sp 1764 from ax4567 27592. Note that ax9 1954 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 1954 . . 3  |-  -.  A. x  -.  x  =  y
2 pm2.21 103 . . . 4  |-  ( -. 
ph  ->  ( ph  ->  A. x ( A. x ph  ->  -.  x  =  y ) ) )
3 ax-1 6 . . . 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 27592 . . . 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 19 . . 3  |-  ( -. 
ph  ->  ( A. x ph  ->  A. x  -.  x  =  y ) )
61, 5mtoi 172 . 2  |-  ( -. 
ph  ->  -.  A. x ph )
76con4i 125 1  |-  ( A. x ph  ->  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1550    = wceq 1653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552
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