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

Step	Hyp	Ref	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)
3	2	con4i 112	. 2 ⊢ (𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
4	1, 3	mpg 1715	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

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