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Theorem ax467 1752
Description: Proof of a single axiom that can replace ax-4 1692, ax-6o 1697, and ax-7 1535 in a subsystem that includes these axioms plus ax-5o 1694 and ax-gen 1536 (and propositional calculus). See ax467to4 1753, ax467to6 1754, and ax467to7 1755 for the re-derivation of those axioms. This theorem extends the idea in Scott Fenton's ax46 1746. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.)
Assertion
Ref Expression
ax467  |-  ( ( A. x A. y  -.  A. x A. y ph  ->  A. x ph )  ->  ph )

Proof of Theorem ax467
StepHypRef Expression
1 ax-4 1692 . . 3  |-  ( A. y ph  ->  ph )
2 hbn1 1564 . . . 4  |-  ( -. 
A. y ph  ->  A. y  -.  A. y ph )
3 ax-6o 1697 . . . . . 6  |-  ( -. 
A. x  -.  A. x A. y ph  ->  A. y ph )
43con1i 123 . . . . 5  |-  ( -. 
A. y ph  ->  A. x  -.  A. x A. y ph )
54alimi 1546 . . . 4  |-  ( A. y  -.  A. y ph  ->  A. y A. x  -.  A. x A. y ph )
6 ax-7 1535 . . . 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 136 . 2  |-  ( -. 
A. x A. y  -.  A. x A. y ph  ->  ph )
9 ax-4 1692 . 2  |-  ( A. x ph  ->  ph )
108, 9ja 155 1  |-  ( ( A. x A. y  -.  A. x A. y ph  ->  A. x ph )  ->  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6   A.wal 1532
This theorem is referenced by:  ax467to4  1753  ax467to6  1754  ax467to7  1755
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-4 1692  ax-6o 1697
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