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Theorem eqneltrd 2717
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 2683 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
41, 3mtbird 315 1 (𝜑 → ¬ 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1480  wcel 1987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-cleq 2614  df-clel 2617
This theorem is referenced by:  eqneltrrd  2718  opabn1stprc  7173  omopth2  7609  fpwwe2  9409  znnn0nn  11433  sqrtneglem  13941  dvdsaddre2b  14953  mreexmrid  16224  mplcoe1  19384  mplcoe5  19387  2sqn0  29428  bj-xnex  32698  islln2a  34280  islpln2a  34311  islvol2aN  34355  oddfl  38950  sumnnodd  39263  sinaover2ne0  39379  dvnprodlem1  39464  dirker2re  39613  dirkerdenne0  39614  dirkertrigeqlem3  39621  dirkercncflem1  39624  dirkercncflem2  39625  dirkercncflem4  39627  fouriersw  39752
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