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Theorem ax6dgen 2160
Description: Tarski's system uses the weaker ax6v 2058 instead of the bundled ax-6 2057, so here we show that the degenerate case of ax-6 2057 can be derived. Even though ax-6 2057 is in the list of axioms used, recall that in set.mm, the only statement referencing ax-6 2057 is ax6v 2058. We later rederive from ax6v 2058 the bundled form as ax6 2413 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 2097 . 2 𝑥 = 𝑥
21notnoti 139 . . 3 ¬ ¬ 𝑥 = 𝑥
32spfalw 2087 . 2 (∀𝑥 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)
41, 3mt2 191 1 ¬ ∀𝑥 ¬ 𝑥 = 𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093
This theorem depends on definitions:  df-bi 197  df-ex 1853
This theorem is referenced by: (None)
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