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Theorem ax-9 1541
Description: Derive ax-9 1541 from ax-i9 1540, the modified version for intuitionistic logic. Although ax-9 1541 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1540. (Contributed by NM, 3-Feb-2015.)
Assertion
Ref Expression
ax-9 ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Proof of Theorem ax-9
StepHypRef Expression
1 ax-i9 1540 . . 3 𝑥 𝑥 = 𝑦
21notnoti 646 . 2 ¬ ¬ ∃𝑥 𝑥 = 𝑦
3 alnex 1509 . 2 (∀𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∃𝑥 𝑥 = 𝑦)
42, 3mtbir 672 1 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wal 1361   = wceq 1363  wex 1502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1457  ax-gen 1459  ax-ie2 1504  ax-i9 1540
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369
This theorem is referenced by:  equidqe  1542
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