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Theorem eqneltrd 2183
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 2156 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
41, 3mtbird 633 1 (𝜑 → ¬ 𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1289  wcel 1438
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-in1 579  ax-in2 580  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-clel 2084
This theorem is referenced by:  iseqf1olemnab  9905  nninfalllemn  11781
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