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Theorem dveeq1 1895
 Description: Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Proof rewritten by Jim Kingdon, 19-Feb-2018.)
Assertion
Ref Expression
dveeq1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Distinct variable group:   𝑥,𝑧

Proof of Theorem dveeq1
StepHypRef Expression
1 dveeq2 1696 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
2 equcom 1593 . 2 (𝑧 = 𝑦𝑦 = 𝑧)
32albii 1359 . 2 (∀𝑥 𝑧 = 𝑦 ↔ ∀𝑥 𝑦 = 𝑧)
41, 2, 33imtr3g 193 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1241 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646 This theorem is referenced by:  sbal2  1898
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