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Theorem dveeq2 2190
Description: Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.) Remove dependency on ax-11 1971. (Revised by Wolf Lammen, 8-Sep-2018.)
Assertion
Ref Expression
dveeq2 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Distinct variable group:   𝑥,𝑧

Proof of Theorem dveeq2
StepHypRef Expression
1 nfeqf2 2189 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
21nfrd 2006 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
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 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-12 1983  ax-13 2137
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-nf 1699
This theorem is referenced by:  axc15  2195  ax12v2OLD  2234
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