Theorem ax6dgen 2165

Description: Tarski's system uses the weaker ax6v 1991 instead of the bundled ax-6 1990, so here we show that the degenerate case of ax-6 1990 can be derived. Even though ax-6 1990 is in the list of axioms used, recall that in set.mm, the only statement referencing ax-6 1990 is ax6v 1991. We later rederive from ax6v 1991 the bundled form as ax6 2418 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 2035	. 2 ⊢ 𝑥 = 𝑥
2	1	notnoti 144	. . 3 ⊢ ¬ ¬ 𝑥 = 𝑥
3	2	spfalw 2003	. 2 ⊢ (∀𝑥 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)
4	1, 3	mt2 203	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑥

Syntax hints:  ¬ wn 3  ∀wal 1561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031

This theorem depends on definitions:  df-bi 210  df-ex 1803

This theorem is referenced by: (None)