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Theorem neleqtrd 2856
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 2820 . 2 (𝜑 → (𝐶𝐴𝐶𝐵))
41, 3mtbid 324 1 (𝜑 → ¬ 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wcel 2107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-cleq 2725  df-clel 2811
This theorem is referenced by:  neleqtrrd  2857  smoord  8312  r1tskina  10723  ofccat  14860  mreexexlem2d  17530  opptgdim2  27729  acopyeu  27818  dochnel  39902  stoweidlem26  44353  fourierdlem60  44493  fourierdlem61  44494  sge00  44703  sge0sn  44706  sge0split  44736
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