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Theorem dveel1 1996
 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 ax-17 1507 . 2 (𝑤𝑧 → ∀𝑥 𝑤𝑧)
2 ax-17 1507 . 2 (𝑦𝑧 → ∀𝑤 𝑦𝑧)
3 elequ1 1691 . 2 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
41, 2, 3dvelimf 1991 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦𝑧 → ∀𝑥 𝑦𝑧))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1330 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737 This theorem is referenced by: (None)
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