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Theorem eqneltrrd 2708
 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.) (Proof shortened by Wolf Lammen, 13-Nov-2019.)
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.1 . . 3 (𝜑𝐴 = 𝐵)
21eqcomd 2616 . 2 (𝜑𝐵 = 𝐴)
3 eqneltrrd.2 . 2 (𝜑 → ¬ 𝐴𝐶)
42, 3eqneltrd 2707 1 (𝜑 → ¬ 𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ∈ wcel 1977 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606 This theorem is referenced by:  bitsf1  15006  lssvancl2  18767  lbsind2  18902  lindfind2  19976  2atjlej  33783  2atnelvolN  33891  lmod1zrnlvec  42077
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