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Theorem ax467 2219
Description: Proof of a single axiom that can replace ax-4 2185, ax-6o 2187, and ax-7 1745 in a subsystem that includes these axioms plus ax-5o 2186 and ax-gen 1552 (and propositional calculus). See ax467to4 2220, ax467to6 2221, and ax467to7 2222 for the re-derivation of those axioms. This theorem extends the idea in Scott Fenton's ax46 2212. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage 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 2185 . . 3  |-  ( A. y ph  ->  ph )
2 ax6 2197 . . . 4  |-  ( -. 
A. y ph  ->  A. y  -.  A. y ph )
3 ax-6o 2187 . . . . . 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 1565 . . . 4  |-  ( A. y  -.  A. y ph  ->  A. y A. x  -.  A. x A. y ph )
6 ax-7 1745 . . . 4  |-  ( A. y A. x  -.  A. x A. y ph  ->  A. x A. y  -. 
A. x A. y ph )
72, 5, 63syl 19 . . 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 2185 . 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 3    -> wi 4   A.wal 1546
This theorem is referenced by:  ax467to4  2220  ax467to6  2221  ax467to7  2222
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-7 1745  ax-4 2185  ax-5o 2186  ax-6o 2187
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