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Theorem ax6fromc10 33199
 Description: Rederivation of axiom ax-6 1875 from ax-c7 33188, ax-c10 33189, ax-gen 1713 and propositional calculus. See axc10 2240 for the derivation of ax-c10 33189 from ax-6 1875. Lemma L18 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax6fromc10 ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Proof of Theorem ax6fromc10
StepHypRef Expression
1 ax-c10 33189 . 2 (∀𝑥(𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 ¬ 𝑥 = 𝑦) → ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
2 ax-c7 33188 . . 3 (¬ ∀𝑥 ¬ ∀𝑥 ¬ 𝑥 = 𝑦 → ¬ 𝑥 = 𝑦)
32con4i 112 . 2 (𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
41, 3mpg 1715 1 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473 This theorem was proved from axioms:  ax-mp 5  ax-3 8  ax-gen 1713  ax-c7 33188  ax-c10 33189 This theorem is referenced by:  equidqe  33225
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