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Theorem dveel1 2516
 Description: Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
Assertion
Ref Expression
dveel1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦𝑧 → ∀𝑥 𝑦𝑧))
Distinct variable group:   𝑥,𝑧

Proof of Theorem dveel1
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elequ1 2151 . 2 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
21dvelimv 2487 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦𝑧 → ∀𝑥 𝑦𝑧))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1628 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857 This theorem is referenced by:  distel  32039
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