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Theorem neleqtrd 2859
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 2823 . 2 (𝜑 → (𝐶𝐴𝐶𝐵))
41, 3mtbid 327 1 (𝜑 → ¬ 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1543  wcel 2110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-cleq 2729  df-clel 2816
This theorem is referenced by:  neleqtrrd  2860  smoord  8102  r1tskina  10396  ofccat  14532  mreexexlem2d  17148  opptgdim2  26836  acopyeu  26925  dochnel  39144  stoweidlem26  43242  fourierdlem60  43382  fourierdlem61  43383  sge00  43589  sge0sn  43592  sge0split  43622
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