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Theorem ax467 2169
 Description: Proof of a single axiom that can replace ax-4 2135, ax-6o 2137, and ax-7 1734 in a subsystem that includes these axioms plus ax-5o 2136 and ax-gen 1546 (and propositional calculus). See ax467to4 2170, ax467to6 2171, and ax467to7 2172 for the re-derivation of those axioms. This theorem extends the idea in Scott Fenton's ax46 2162. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax467 ((xy ¬ xyφxφ) → φ)

Proof of Theorem ax467
StepHypRef Expression
1 ax-4 2135 . . 3 (yφφ)
2 ax6 2147 . . . 4 yφy ¬ yφ)
3 ax-6o 2137 . . . . . 6 x ¬ xyφyφ)
43con1i 121 . . . . 5 yφx ¬ xyφ)
54alimi 1559 . . . 4 (y ¬ yφyx ¬ xyφ)
6 ax-7 1734 . . . 4 (yx ¬ xyφxy ¬ xyφ)
72, 5, 63syl 18 . . 3 yφxy ¬ xyφ)
81, 7nsyl4 134 . 2 xy ¬ xyφφ)
9 ax-4 2135 . 2 (xφφ)
108, 9ja 153 1 ((xy ¬ xyφxφ) → φ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-7 1734  ax-4 2135  ax-5o 2136  ax-6o 2137 This theorem is referenced by:  ax467to4  2170  ax467to6  2171  ax467to7  2172
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