New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  inssdif0 GIF version

Theorem inssdif0 3617
 Description: Intersection, subclass, and difference relationship. (Contributed by NM, 27-Oct-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
Assertion
Ref Expression
inssdif0 ((AB) C ↔ (A ∩ (B C)) = )

Proof of Theorem inssdif0
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elin 3219 . . . . . 6 (x (AB) ↔ (x A x B))
21imbi1i 315 . . . . 5 ((x (AB) → x C) ↔ ((x A x B) → x C))
3 iman 413 . . . . 5 (((x A x B) → x C) ↔ ¬ ((x A x B) ¬ x C))
42, 3bitri 240 . . . 4 ((x (AB) → x C) ↔ ¬ ((x A x B) ¬ x C))
5 eldif 3221 . . . . . 6 (x (B C) ↔ (x B ¬ x C))
65anbi2i 675 . . . . 5 ((x A x (B C)) ↔ (x A (x B ¬ x C)))
7 elin 3219 . . . . 5 (x (A ∩ (B C)) ↔ (x A x (B C)))
8 anass 630 . . . . 5 (((x A x B) ¬ x C) ↔ (x A (x B ¬ x C)))
96, 7, 83bitr4ri 269 . . . 4 (((x A x B) ¬ x C) ↔ x (A ∩ (B C)))
104, 9xchbinx 301 . . 3 ((x (AB) → x C) ↔ ¬ x (A ∩ (B C)))
1110albii 1566 . 2 (x(x (AB) → x C) ↔ x ¬ x (A ∩ (B C)))
12 dfss2 3262 . 2 ((AB) Cx(x (AB) → x C))
13 eq0 3564 . 2 ((A ∩ (B C)) = x ¬ x (A ∩ (B C)))
1411, 12, 133bitr4i 268 1 ((AB) C ↔ (A ∩ (B C)) = )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710   ∖ cdif 3206   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551 This theorem is referenced by:  disjdif  3622
 Copyright terms: Public domain W3C validator