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

Theorem indif 3350
 Description: Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
indif (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)

Proof of Theorem indif
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 anabs5 563 . . 3 ((𝑥𝐴 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
2 elin 3290 . . . 4 (𝑥 ∈ (𝐴 ∩ (𝐴𝐵)) ↔ (𝑥𝐴𝑥 ∈ (𝐴𝐵)))
3 eldif 3111 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
43anbi2i 453 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐴𝐵)) ↔ (𝑥𝐴 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)))
52, 4bitri 183 . . 3 (𝑥 ∈ (𝐴 ∩ (𝐴𝐵)) ↔ (𝑥𝐴 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)))
61, 5, 33bitr4i 211 . 2 (𝑥 ∈ (𝐴 ∩ (𝐴𝐵)) ↔ 𝑥 ∈ (𝐴𝐵))
76eqriv 2154 1 (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 103   = wceq 1335   ∈ wcel 2128   ∖ cdif 3099   ∩ cin 3101 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-in 3108 This theorem is referenced by:  resdif  5433
 Copyright terms: Public domain W3C validator