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Theorem dveel2 2442
Description: Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
Assertion
Ref Expression
dveel2 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧𝑦 → ∀𝑥 𝑧𝑦))
Distinct variable group:   𝑥,𝑧

Proof of Theorem dveel2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elequ2 2096 . 2 (𝑤 = 𝑦 → (𝑧𝑤𝑧𝑦))
21dvelimv 2431 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧𝑦 → ∀𝑥 𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766
This theorem is referenced by:  axc14  2443
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