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Theorem dveel2 1939
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 ax-17 1460 . 2 (𝑧𝑤 → ∀𝑥 𝑧𝑤)
2 ax-17 1460 . 2 (𝑧𝑦 → ∀𝑤 𝑧𝑦)
3 elequ2 1642 . 2 (𝑤 = 𝑦 → (𝑧𝑤𝑧𝑦))
41, 2, 3dvelimf 1933 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧𝑦 → ∀𝑥 𝑧𝑦))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687
This theorem is referenced by: (None)
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