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Theorem eqneltrrd 2237
 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 2209 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
41, 3mtbid 662 1 (𝜑 → ¬ 𝐵𝐶)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1332   ∈ wcel 1481 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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-cleq 2133  df-clel 2136 This theorem is referenced by:  ctinf  11972
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