Theorem ax6fromc10 37754

Description: Rederivation of Axiom ax-6 1971 from ax-c7 37743, ax-c10 37744, ax-gen 1797 and propositional calculus. See axc10 2384 for the derivation of ax-c10 37744 from ax-6 1971. Lemma L18 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) Use ax-6 1971 instead. (New usage is discouraged.)

Assertion

Ref	Expression
ax6fromc10	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Proof of Theorem ax6fromc10

Step	Hyp	Ref	Expression
1		ax-c10 37744	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 ¬ 𝑥 = 𝑦) → ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
2		ax-c7 37743	. . 3 ⊢ (¬ ∀𝑥 ¬ ∀𝑥 ¬ 𝑥 = 𝑦 → ¬ 𝑥 = 𝑦)
3	2	con4i 114	. 2 ⊢ (𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
4	1, 3	mpg 1799	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-3 8  ax-gen 1797  ax-c7 37743  ax-c10 37744

This theorem is referenced by:  equidqe  37780