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Theorem neleqtrd 2176
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 2148 . 2 (𝜑 → (𝐶𝐴𝐶𝐵))
41, 3mtbid 629 1 (𝜑 → ¬ 𝐶𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1284  wcel 1433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063
This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077
This theorem is referenced by: (None)
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