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Theorem difundi 3387
Description: Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
difundi (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))

Proof of Theorem difundi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eldif 3138 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
2 eldif 3138 . . . 4 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐶))
31, 2anbi12i 460 . . 3 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥 ∈ (𝐴𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
4 elin 3318 . . 3 (𝑥 ∈ ((𝐴𝐵) ∩ (𝐴𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝑥 ∈ (𝐴𝐶)))
5 eldif 3138 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐶)))
6 elun 3276 . . . . . . . 8 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
76notbii 668 . . . . . . 7 𝑥 ∈ (𝐵𝐶) ↔ ¬ (𝑥𝐵𝑥𝐶))
87anbi2i 457 . . . . . 6 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
95, 8bitri 184 . . . . 5 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
10 ioran 752 . . . . . 6 (¬ (𝑥𝐵𝑥𝐶) ↔ (¬ 𝑥𝐵 ∧ ¬ 𝑥𝐶))
1110anbi2i 457 . . . . 5 ((𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵 ∧ ¬ 𝑥𝐶)))
129, 11bitri 184 . . . 4 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵 ∧ ¬ 𝑥𝐶)))
13 anandi 590 . . . 4 ((𝑥𝐴 ∧ (¬ 𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
1412, 13bitri 184 . . 3 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
153, 4, 143bitr4ri 213 . 2 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∩ (𝐴𝐶)))
1615eqriv 2174 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104  wo 708   = wceq 1353  wcel 2148  cdif 3126  cun 3127  cin 3128
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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131  df-un 3133  df-in 3135
This theorem is referenced by:  undm  3393  undifdc  6917  uncld  13280
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