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Theorem eqneltrrd 2134
 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
eqneltrrd.1 (𝜑𝐴 = 𝐵)
eqneltrrd.2 (𝜑 → ¬ 𝐴𝐶)
Assertion
Ref Expression
eqneltrrd (𝜑 → ¬ 𝐵𝐶)

Proof of Theorem eqneltrrd
StepHypRef Expression
1 eqneltrrd.2 . 2 (𝜑 → ¬ 𝐴𝐶)
2 eqneltrrd.1 . . 3 (𝜑𝐴 = 𝐵)
32eleq1d 2106 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
41, 3mtbid 597 1 (𝜑 → ¬ 𝐵𝐶)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1243   ∈ wcel 1393 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-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036 This theorem is referenced by: (None)
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