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Theorem difundi 3166
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 2904 . . . 4 (x (AB) ↔ (x A ¬ x B))
2 eldif 2904 . . . 4 (x (A𝐶) ↔ (x A ¬ x 𝐶))
31, 2anbi12i 436 . . 3 ((x (AB) x (A𝐶)) ↔ ((x A ¬ x B) (x A ¬ x 𝐶)))
4 elin 3103 . . 3 (x ((AB) ∩ (A𝐶)) ↔ (x (AB) x (A𝐶)))
5 eldif 2904 . . . . . 6 (x (A ∖ (B𝐶)) ↔ (x A ¬ x (B𝐶)))
6 elun 3061 . . . . . . . 8 (x (B𝐶) ↔ (x B x 𝐶))
76notbii 581 . . . . . . 7 x (B𝐶) ↔ ¬ (x B x 𝐶))
87anbi2i 433 . . . . . 6 ((x A ¬ x (B𝐶)) ↔ (x A ¬ (x B x 𝐶)))
95, 8bitri 173 . . . . 5 (x (A ∖ (B𝐶)) ↔ (x A ¬ (x B x 𝐶)))
10 ioran 656 . . . . . 6 (¬ (x B x 𝐶) ↔ (¬ x B ¬ x 𝐶))
1110anbi2i 433 . . . . 5 ((x A ¬ (x B x 𝐶)) ↔ (x A x B ¬ x 𝐶)))
129, 11bitri 173 . . . 4 (x (A ∖ (B𝐶)) ↔ (x A x B ¬ x 𝐶)))
13 anandi 511 . . . 4 ((x A x B ¬ x 𝐶)) ↔ ((x A ¬ x B) (x A ¬ x 𝐶)))
1412, 13bitri 173 . . 3 (x (A ∖ (B𝐶)) ↔ ((x A ¬ x B) (x A ¬ x 𝐶)))
153, 4, 143bitr4ri 202 . 2 (x (A ∖ (B𝐶)) ↔ x ((AB) ∩ (A𝐶)))
1615eqriv 2019 1 (A ∖ (B𝐶)) = ((AB) ∩ (A𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   wo 616   = wceq 1228   wcel 1374  cdif 2891  cun 2892  cin 2893
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-dif 2897  df-un 2899  df-in 2901
This theorem is referenced by:  undm  3172
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