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Theorem ax6 2147
 Description: Rederivation of axiom ax-6 1729 from ax-6o 2137 and other older axioms. See ax6o 1750 for the derivation of ax-6o 2137 from ax-6 1729. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax6 xφx ¬ xφ)

Proof of Theorem ax6
StepHypRef Expression
1 ax-5o 2136 . . 3 (x(x ¬ xxφ → ¬ xφ) → (x ¬ xxφx ¬ xφ))
2 ax-4 2135 . . . 4 (x ¬ xxφ → ¬ xxφ)
3 ax-5o 2136 . . . . 5 (x(xφxφ) → (xφxxφ))
4 id 19 . . . . 5 (xφxφ)
53, 4mpg 1548 . . . 4 (xφxxφ)
62, 5nsyl 113 . . 3 (x ¬ xxφ → ¬ xφ)
71, 6mpg 1548 . 2 (x ¬ xxφx ¬ xφ)
8 ax-6o 2137 . 2 x ¬ xxφxφ)
97, 8nsyl4 134 1 xφx ¬ xφ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-4 2135  ax-5o 2136  ax-6o 2137 This theorem is referenced by:  hba1-o  2149  ax467  2169  equidq  2175
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