New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  difin2 GIF version

Theorem difin2 3516
 Description: Represent a set difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
difin2 (A C → (A B) = ((C B) ∩ A))

Proof of Theorem difin2
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 ssel 3267 . . . . 5 (A C → (x Ax C))
21pm4.71d 615 . . . 4 (A C → (x A ↔ (x A x C)))
32anbi1d 685 . . 3 (A C → ((x A ¬ x B) ↔ ((x A x C) ¬ x B)))
4 eldif 3221 . . 3 (x (A B) ↔ (x A ¬ x B))
5 elin 3219 . . . 4 (x ((C B) ∩ A) ↔ (x (C B) x A))
6 eldif 3221 . . . . 5 (x (C B) ↔ (x C ¬ x B))
76anbi1i 676 . . . 4 ((x (C B) x A) ↔ ((x C ¬ x B) x A))
8 ancom 437 . . . . 5 (((x C ¬ x B) x A) ↔ (x A (x C ¬ x B)))
9 anass 630 . . . . 5 (((x A x C) ¬ x B) ↔ (x A (x C ¬ x B)))
108, 9bitr4i 243 . . . 4 (((x C ¬ x B) x A) ↔ ((x A x C) ¬ x B))
115, 7, 103bitri 262 . . 3 (x ((C B) ∩ A) ↔ ((x A x C) ¬ x B))
123, 4, 113bitr4g 279 . 2 (A C → (x (A B) ↔ x ((C B) ∩ A)))
1312eqrdv 2351 1 (A C → (A B) = ((C B) ∩ A))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∖ cdif 3206   ∩ cin 3208   ⊆ wss 3257 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator