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Theorem nd5 1746
 Description: A lemma for proving conditionless ZFC axioms. (Contributed by NM, 8-Jan-2002.)
Assertion
Ref Expression
nd5 (¬ ∀𝑦 𝑦 = 𝑥 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Distinct variable group:   𝑥,𝑧

Proof of Theorem nd5
StepHypRef Expression
1 dveeq2 1743 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
21nalequcoms 1455 1 (¬ ∀𝑦 𝑦 = 𝑥 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1287   = wceq 1289 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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472 This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693 This theorem is referenced by: (None)
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