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Theorem neleqtrd 2911
 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 2875 . 2 (𝜑 → (𝐶𝐴𝐶𝐵))
41, 3mtbid 327 1 (𝜑 → ¬ 𝐶𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538   ∈ wcel 2111 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-clel 2870 This theorem is referenced by:  neleqtrrd  2912  smoord  7988  r1tskina  10196  ofccat  14323  mreexexlem2d  16911  opptgdim2  26549  acopyeu  26638  dochnel  38708  stoweidlem26  42711  fourierdlem60  42851  fourierdlem61  42852  sge00  43058  sge0sn  43061  sge0split  43091
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