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Theorem axc5c711 31921
Description: Proof of a single axiom that can replace ax-c5 31887, ax-c7 31889, and ax-11 1866 in a subsystem that includes these axioms plus ax-c4 31888 and ax-gen 1639 (and propositional calculus). See axc5c711toc5 31922, axc5c711toc7 31923, and axc5c711to11 31924 for the re-derivation of those axioms. This theorem extends the idea in Scott Fenton's axc5c7 31914. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc5c711  |-  ( ( A. x A. y  -.  A. x A. y ph  ->  A. x ph )  ->  ph )

Proof of Theorem axc5c711
StepHypRef Expression
1 ax-c5 31887 . . 3  |-  ( A. y ph  ->  ph )
2 ax10 31899 . . . 4  |-  ( -. 
A. y ph  ->  A. y  -.  A. y ph )
3 ax-c7 31889 . . . . . 6  |-  ( -. 
A. x  -.  A. x A. y ph  ->  A. y ph )
43con1i 129 . . . . 5  |-  ( -. 
A. y ph  ->  A. x  -.  A. x A. y ph )
54alimi 1654 . . . 4  |-  ( A. y  -.  A. y ph  ->  A. y A. x  -.  A. x A. y ph )
6 ax-11 1866 . . . 4  |-  ( A. y A. x  -.  A. x A. y ph  ->  A. x A. y  -. 
A. x A. y ph )
72, 5, 63syl 20 . . 3  |-  ( -. 
A. y ph  ->  A. x A. y  -. 
A. x A. y ph )
81, 7nsyl4 142 . 2  |-  ( -. 
A. x A. y  -.  A. x A. y ph  ->  ph )
9 ax-c5 31887 . 2  |-  ( A. x ph  ->  ph )
108, 9ja 161 1  |-  ( ( A. x A. y  -.  A. x A. y ph  ->  A. x ph )  ->  ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   A.wal 1403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-11 1866  ax-c5 31887  ax-c4 31888  ax-c7 31889
This theorem is referenced by:  axc5c711toc5  31922  axc5c711toc7  31923  axc5c711to11  31924
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