Theorem ax6dgen 2123

Description: Tarski's system uses the weaker ax6v 1971 instead of the bundled ax-6 1970, so here we show that the degenerate case of ax-6 1970 can be derived. Even though ax-6 1970 is in the list of axioms used, recall that in set.mm, the only statement referencing ax-6 1970 is ax6v 1971. We later rederive from ax6v 1971 the bundled form as ax6 2382 with the help of the auxiliary axiom schemes. (Contributed by NM, 23-Apr-2017.)

Assertion

Ref	Expression
ax6dgen	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑥

Proof of Theorem ax6dgen

Step	Hyp	Ref	Expression
1		equid 2014	. 2 ⊢ 𝑥 = 𝑥
2	1	notnoti 143	. . 3 ⊢ ¬ ¬ 𝑥 = 𝑥
3	2	spfalw 2000	. 2 ⊢ (∀𝑥 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)
4	1, 3	mt2 199	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑥

Syntax hints:  ¬ wn 3  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010

This theorem depends on definitions:  df-bi 206  df-ex 1781

This theorem is referenced by: (None)