ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  indifdir Structured version   GIF version

Theorem indifdir 3166
Description: Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)
Assertion
Ref Expression
indifdir ((AB) ∩ 𝐶) = ((A𝐶) ∖ (B𝐶))

Proof of Theorem indifdir
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elin 3099 . . . 4 (x (A𝐶) ↔ (x A x 𝐶))
2 elin 3099 . . . . 5 (x (B𝐶) ↔ (x B x 𝐶))
32notbii 581 . . . 4 x (B𝐶) ↔ ¬ (x B x 𝐶))
41, 3anbi12i 436 . . 3 ((x (A𝐶) ¬ x (B𝐶)) ↔ ((x A x 𝐶) ¬ (x B x 𝐶)))
5 eldif 2900 . . 3 (x ((A𝐶) ∖ (B𝐶)) ↔ (x (A𝐶) ¬ x (B𝐶)))
6 elin 3099 . . . . 5 (x ((AB) ∩ 𝐶) ↔ (x (AB) x 𝐶))
7 eldif 2900 . . . . . 6 (x (AB) ↔ (x A ¬ x B))
87anbi1i 434 . . . . 5 ((x (AB) x 𝐶) ↔ ((x A ¬ x B) x 𝐶))
96, 8bitri 173 . . . 4 (x ((AB) ∩ 𝐶) ↔ ((x A ¬ x B) x 𝐶))
10 an32 484 . . . . 5 (((x A ¬ x B) x 𝐶) ↔ ((x A x 𝐶) ¬ x B))
11 ax-ia1 99 . . . . . . . 8 ((x B x 𝐶) → x B)
1211con3i 549 . . . . . . 7 x B → ¬ (x B x 𝐶))
1312anim2i 324 . . . . . 6 (((x A x 𝐶) ¬ x B) → ((x A x 𝐶) ¬ (x B x 𝐶)))
14 ax-ia1 99 . . . . . . 7 (((x A x 𝐶) ¬ (x B x 𝐶)) → (x A x 𝐶))
15 ax-in2 533 . . . . . . . . . . 11 (¬ (x B x 𝐶) → ((x B x 𝐶) → ⊥ ))
1615expcomd 1306 . . . . . . . . . 10 (¬ (x B x 𝐶) → (x 𝐶 → (x B → ⊥ )))
1716impcom 116 . . . . . . . . 9 ((x 𝐶 ¬ (x B x 𝐶)) → (x B → ⊥ ))
18 dfnot 1245 . . . . . . . . 9 x B ↔ (x B → ⊥ ))
1917, 18sylibr 137 . . . . . . . 8 ((x 𝐶 ¬ (x B x 𝐶)) → ¬ x B)
2019adantll 448 . . . . . . 7 (((x A x 𝐶) ¬ (x B x 𝐶)) → ¬ x B)
2114, 20jca 290 . . . . . 6 (((x A x 𝐶) ¬ (x B x 𝐶)) → ((x A x 𝐶) ¬ x B))
2213, 21impbii 117 . . . . 5 (((x A x 𝐶) ¬ x B) ↔ ((x A x 𝐶) ¬ (x B x 𝐶)))
2310, 22bitri 173 . . . 4 (((x A ¬ x B) x 𝐶) ↔ ((x A x 𝐶) ¬ (x B x 𝐶)))
249, 23bitri 173 . . 3 (x ((AB) ∩ 𝐶) ↔ ((x A x 𝐶) ¬ (x B x 𝐶)))
254, 5, 243bitr4ri 202 . 2 (x ((AB) ∩ 𝐶) ↔ x ((A𝐶) ∖ (B𝐶)))
2625eqriv 2015 1 ((AB) ∩ 𝐶) = ((A𝐶) ∖ (B𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   = wceq 1226  wfal 1231   wcel 1370  cdif 2887  cin 2889
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-dif 2893  df-in 2897
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator