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Theorem neleqtrd 2709
 Description: If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
neleqtrd.1 (𝜑 → ¬ 𝐶𝐴)
neleqtrd.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
neleqtrd (𝜑 → ¬ 𝐶𝐵)

Proof of Theorem neleqtrd
StepHypRef Expression
1 neleqtrd.1 . 2 (𝜑 → ¬ 𝐶𝐴)
2 neleqtrd.2 . . 3 (𝜑𝐴 = 𝐵)
32eleq2d 2673 . 2 (𝜑 → (𝐶𝐴𝐶𝐵))
41, 3mtbid 313 1 (𝜑 → ¬ 𝐶𝐵)
 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-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:  neleqtrrd  2710  smoord  7349  r1tskina  9483  ofccat  13556  mreexexlem2d  16128  opptgdim2  25437  dochnel  35700  stoweidlem26  38919  fourierdlem60  39059  fourierdlem61  39060  sge00  39269  sge0sn  39272  sge0split  39302  iundjiunlem  39352
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