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Theorem eqneltrrd 2274
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 → ¬ 𝐴𝐶)
Assertion
Ref Expression
eqneltrrd (𝜑 → ¬ 𝐵𝐶)

Proof of Theorem eqneltrrd
StepHypRef Expression
1 eqneltrrd.2 . 2 (𝜑 → ¬ 𝐴𝐶)
2 eqneltrrd.1 . . 3 (𝜑𝐴 = 𝐵)
32eleq1d 2246 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
41, 3mtbid 672 1 (𝜑 → ¬ 𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1353  wcel 2148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173
This theorem is referenced by:  ctinf  12411
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