Metamath Proof Explorer


Theorem ax6fromc10

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

Ref Expression
Assertion ax6fromc10 ¬ ∀ 𝑥 ¬ 𝑥 = 𝑦

Proof

Step Hyp Ref Expression
1 ax-c10 ( ∀ 𝑥 ( 𝑥 = 𝑦 → ∀ 𝑥 ¬ ∀ 𝑥 ¬ 𝑥 = 𝑦 ) → ¬ ∀ 𝑥 ¬ 𝑥 = 𝑦 )
2 ax-c7 ( ¬ ∀ 𝑥 ¬ ∀ 𝑥 ¬ 𝑥 = 𝑦 → ¬ 𝑥 = 𝑦 )
3 2 con4i ( 𝑥 = 𝑦 → ∀ 𝑥 ¬ ∀ 𝑥 ¬ 𝑥 = 𝑦 )
4 1 3 mpg ¬ ∀ 𝑥 ¬ 𝑥 = 𝑦