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Theorem dveeq1 2283
 Description: Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) Remove dependency on ax-11 2019. (Revised by Wolf Lammen, 8-Sep-2018.)
Assertion
Ref Expression
dveeq1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Distinct variable group:   𝑥,𝑧

Proof of Theorem dveeq1
StepHypRef Expression
1 nfeqf1 2282 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
21nf5rd 2051 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1472 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-12 2031  ax-13 2228 This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700 This theorem is referenced by:  nfeqf  2284  axc11nlemALT  2289  axc11n  2290  axc11nOLD  2291  axc11nOLDOLD  2292  axc11nALT  2293
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