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Theorem axc5c4c711toc5 36605
Description: Re-derivation of sp 1909 from axc5c4c711 36604. Note that ax6 2056 is used for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) Revised to use ax6v 1795 instead of ax6 2056, so that this re-derivation requires only ax6v 1795 and propositional calculus. (Revised by BJ, 14-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc5c4c711toc5  |-  ( A. x ph  ->  ph )

Proof of Theorem axc5c4c711toc5
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ax6v 1795 . . 3  |-  -.  A. x  -.  x  =  y
2 pm2.21 111 . . . 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 axc5c4c711 36604 . . . 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 181 . 2  |-  ( -. 
ph  ->  -.  A. x ph )
76con4i 133 1  |-  ( A. x ph  ->  ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1435    = wceq 1437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904
This theorem depends on definitions:  df-bi 188  df-ex 1660
This theorem is referenced by: (None)
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