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Theorem indif 3289
Description: Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
indif (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)

Proof of Theorem indif
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 anabs5 547 . . 3 ((𝑥𝐴 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
2 elin 3229 . . . 4 (𝑥 ∈ (𝐴 ∩ (𝐴𝐵)) ↔ (𝑥𝐴𝑥 ∈ (𝐴𝐵)))
3 eldif 3050 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
43anbi2i 452 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐴𝐵)) ↔ (𝑥𝐴 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)))
52, 4bitri 183 . . 3 (𝑥 ∈ (𝐴 ∩ (𝐴𝐵)) ↔ (𝑥𝐴 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)))
61, 5, 33bitr4i 211 . 2 (𝑥 ∈ (𝐴 ∩ (𝐴𝐵)) ↔ 𝑥 ∈ (𝐴𝐵))
76eqriv 2114 1 (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103   = wceq 1316  wcel 1465  cdif 3038  cin 3040
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-in 3047
This theorem is referenced by:  resdif  5357
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