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Theorem difundi 3183
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 2921 . . . 4 (x (AB) ↔ (x A ¬ x B))
2 eldif 2921 . . . 4 (x (A𝐶) ↔ (x A ¬ x 𝐶))
31, 2anbi12i 433 . . 3 ((x (AB) x (A𝐶)) ↔ ((x A ¬ x B) (x A ¬ x 𝐶)))
4 elin 3120 . . 3 (x ((AB) ∩ (A𝐶)) ↔ (x (AB) x (A𝐶)))
5 eldif 2921 . . . . . 6 (x (A ∖ (B𝐶)) ↔ (x A ¬ x (B𝐶)))
6 elun 3078 . . . . . . . 8 (x (B𝐶) ↔ (x B x 𝐶))
76notbii 593 . . . . . . 7 x (B𝐶) ↔ ¬ (x B x 𝐶))
87anbi2i 430 . . . . . 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 668 . . . . . 6 (¬ (x B x 𝐶) ↔ (¬ x B ¬ x 𝐶))
1110anbi2i 430 . . . . 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 524 . . . 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 2034 1 (A ∖ (B𝐶)) = ((AB) ∩ (A𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   wo 628   = wceq 1242   wcel 1390  cdif 2908  cun 2909  cin 2910
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-un 2916  df-in 2918
This theorem is referenced by:  undm  3189
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