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Theorem eqneltrrd 2447
 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
eqneltrrd.1 (φA = B)
eqneltrrd.2 (φ → ¬ A C)
Assertion
Ref Expression
eqneltrrd (φ → ¬ B C)

Proof of Theorem eqneltrrd
StepHypRef Expression
1 eqneltrrd.2 . 2 (φ → ¬ A C)
2 eqneltrrd.1 . . 3 (φA = B)
32eleq1d 2419 . 2 (φ → (A CB C))
41, 3mtbid 291 1 (φ → ¬ B C)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642   ∈ wcel 1710 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349 This theorem is referenced by: (None)
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