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Theorem eqneltrd 2253
 Description: If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
eqneltrd.1 (𝜑𝐴 = 𝐵)
eqneltrd.2 (𝜑 → ¬ 𝐵𝐶)
Assertion
Ref Expression
eqneltrd (𝜑 → ¬ 𝐴𝐶)

Proof of Theorem eqneltrd
StepHypRef Expression
1 eqneltrd.2 . 2 (𝜑 → ¬ 𝐵𝐶)
2 eqneltrd.1 . . 3 (𝜑𝐴 = 𝐵)
32eleq1d 2226 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
41, 3mtbird 663 1 (𝜑 → ¬ 𝐴𝐶)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1335   ∈ wcel 2128 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-in1 604  ax-in2 605  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-cleq 2150  df-clel 2153 This theorem is referenced by:  iseqf1olemnab  10369  fprodunsn  11483  ctinfomlemom  12128  nninfalllemn  13542
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