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Theorem eldifbd 2924
Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 2921. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
eldifbd.1 (φA (B𝐶))
Assertion
Ref Expression
eldifbd (φ → ¬ A 𝐶)

Proof of Theorem eldifbd
StepHypRef Expression
1 eldifbd.1 . . 3 (φA (B𝐶))
2 eldif 2921 . . 3 (A (B𝐶) ↔ (A B ¬ A 𝐶))
31, 2sylib 127 . 2 (φ → (A B ¬ A 𝐶))
43simprd 107 1 (φ → ¬ A 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   wcel 1390  cdif 2908
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914
This theorem is referenced by: (None)
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