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Theorem neleqtrrd 2118
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
neleqtrrd.1 (φ → ¬ 𝐶 B)
neleqtrrd.2 (φA = B)
Assertion
Ref Expression
neleqtrrd (φ → ¬ 𝐶 A)

Proof of Theorem neleqtrrd
StepHypRef Expression
1 neleqtrrd.1 . 2 (φ → ¬ 𝐶 B)
2 neleqtrrd.2 . . 3 (φA = B)
32eleq2d 2089 . 2 (φ → (𝐶 A𝐶 B))
41, 3mtbird 585 1 (φ → ¬ 𝐶 A)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1228   wcel 1374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-ial 1409  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-cleq 2015  df-clel 2018
This theorem is referenced by: (None)
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