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Theorem ax6dgen 2045
 Description: Tarski's system uses the weaker ax6v 1946 instead of the bundled ax-6 1945, so here we show that the degenerate case of ax-6 1945 can be derived. Even though ax-6 1945 is in the list of axioms used, recall that in set.mm, the only statement referencing ax-6 1945 is ax6v 1946. We later rederive from ax6v 1946 the bundled form as ax6 2287 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 1985 . 2 𝑥 = 𝑥
21notnoti 137 . . 3 ¬ ¬ 𝑥 = 𝑥
32spfalw 1975 . 2 (∀𝑥 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)
41, 3mt2 191 1 ¬ ∀𝑥 ¬ 𝑥 = 𝑥
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3  ∀wal 1521 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981 This theorem depends on definitions:  df-bi 197  df-ex 1745 This theorem is referenced by: (None)
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