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Theorem eqneltrrd 2718
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 2627 . 2 (𝜑𝐵 = 𝐴)
3 eqneltrrd.2 . 2 (𝜑 → ¬ 𝐴𝐶)
42, 3eqneltrd 2717 1 (𝜑 → ¬ 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1480  wcel 1987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-cleq 2614  df-clel 2617
This theorem is referenced by:  bitsf1  15103  lssvancl2  18878  lbsind2  19013  lindfind2  20089  2atjlej  34280  2atnelvolN  34388  lmod1zrnlvec  41597
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