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Theorem ax6dgen 1953
Description: Tarski's system uses the weaker ax6v 1839 instead of the bundled ax-6 1838, so here we show that the degenerate case of ax-6 1838 can be derived. Even though ax-6 1838 is in the list of axioms used, recall that in set.mm, the only statement referencing ax-6 1838 is ax6v 1839. We later rederive from ax6v 1839 the bundled form as ax6 2142 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 1889 . 2 𝑥 = 𝑥
21notnoti 135 . . 3 ¬ ¬ 𝑥 = 𝑥
32spfalw 1879 . 2 (∀𝑥 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)
41, 3mt2 189 1 ¬ ∀𝑥 ¬ 𝑥 = 𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885
This theorem depends on definitions:  df-bi 195  df-ex 1695
This theorem is referenced by: (None)
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