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Theorem difundi 3216
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 2954 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
2 eldif 2954 . . . 4 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐶))
31, 2anbi12i 441 . . 3 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥 ∈ (𝐴𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
4 elin 3153 . . 3 (𝑥 ∈ ((𝐴𝐵) ∩ (𝐴𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝑥 ∈ (𝐴𝐶)))
5 eldif 2954 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐶)))
6 elun 3111 . . . . . . . 8 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
76notbii 604 . . . . . . 7 𝑥 ∈ (𝐵𝐶) ↔ ¬ (𝑥𝐵𝑥𝐶))
87anbi2i 438 . . . . . 6 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
95, 8bitri 177 . . . . 5 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
10 ioran 679 . . . . . 6 (¬ (𝑥𝐵𝑥𝐶) ↔ (¬ 𝑥𝐵 ∧ ¬ 𝑥𝐶))
1110anbi2i 438 . . . . 5 ((𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵 ∧ ¬ 𝑥𝐶)))
129, 11bitri 177 . . . 4 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵 ∧ ¬ 𝑥𝐶)))
13 anandi 532 . . . 4 ((𝑥𝐴 ∧ (¬ 𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
1412, 13bitri 177 . . 3 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
153, 4, 143bitr4ri 206 . 2 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∩ (𝐴𝐶)))
1615eqriv 2053 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 101  wo 639   = wceq 1259  wcel 1409  cdif 2941  cun 2942  cin 2943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2947  df-un 2949  df-in 2951
This theorem is referenced by:  undm  3222
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