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Theorem indif 3283
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 545 . . 3 ((𝑥𝐴 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
2 elin 3223 . . . 4 (𝑥 ∈ (𝐴 ∩ (𝐴𝐵)) ↔ (𝑥𝐴𝑥 ∈ (𝐴𝐵)))
3 eldif 3044 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
43anbi2i 450 . . . 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 2110 1 (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103   = wceq 1312  wcel 1461  cdif 3032  cin 3034
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-dif 3037  df-in 3041
This theorem is referenced by:  resdif  5343
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