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Theorem ax6dgen 2129
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 2391 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
StepHypRef Expression
1 equid 2019 . 2 𝑥 = 𝑥
21notnoti 145 . . 3 ¬ ¬ 𝑥 = 𝑥
32spfalw 2004 . 2 (∀𝑥 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)
41, 3mt2 203 1 ¬ ∀𝑥 ¬ 𝑥 = 𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015
This theorem depends on definitions:  df-bi 210  df-ex 1782
This theorem is referenced by: (None)
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