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Theorem eqneltrd 2149
 Description: If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
eqneltrd.1 (𝜑𝐴 = 𝐵)
eqneltrd.2 (𝜑 → ¬ 𝐵𝐶)
Assertion
Ref Expression
eqneltrd (𝜑 → ¬ 𝐴𝐶)

Proof of Theorem eqneltrd
StepHypRef Expression
1 eqneltrd.2 . 2 (𝜑 → ¬ 𝐵𝐶)
2 eqneltrd.1 . . 3 (𝜑𝐴 = 𝐵)
32eleq1d 2122 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
41, 3mtbird 608 1 (𝜑 → ¬ 𝐴𝐶)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1259   ∈ wcel 1409 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-clel 2052 This theorem is referenced by: (None)
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