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Theorem dveel1 2484
 Description: Quantifier introduction when one pair of variables is distinct. Usage of this theorem is discouraged because it depends on ax-13 2390. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)
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 2121 . 2 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
21dvelimv 2474 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦𝑧 → ∀𝑥 𝑦𝑧))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1535 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785 This theorem is referenced by:  distel  33056
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