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Theorem ax-9 1465
 Description: Derive ax-9 1465 from ax-i9 1464, the modified version for intuitionistic logic. Although ax-9 1465 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1464. (Contributed by NM, 3-Feb-2015.)
Assertion
Ref Expression
ax-9 ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Proof of Theorem ax-9
StepHypRef Expression
1 ax-i9 1464 . . 3 𝑥 𝑥 = 𝑦
21notnoti 607 . 2 ¬ ¬ ∃𝑥 𝑥 = 𝑦
3 alnex 1429 . 2 (∀𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∃𝑥 𝑥 = 𝑦)
42, 3mtbir 629 1 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
 Colors of variables: wff set class Syntax hints:  ¬ wn 3  ∀wal 1283   = wceq 1285  ∃wex 1422 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie2 1424  ax-i9 1464 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291 This theorem is referenced by:  equidqe  1466
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