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Theorem eqneltrd 2211
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 2184 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
41, 3mtbird 645 1 (𝜑 → ¬ 𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1314  wcel 1463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-cleq 2108  df-clel 2111
This theorem is referenced by:  iseqf1olemnab  10201  ctinfomlemom  11835  nninfalllemn  13004
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