Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfin2 Unicode version

Theorem dfin2 3733
 Description: An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 3732. Another version is given by dfin4 3737. (Contributed by NM, 10-Jun-2004.)
Assertion
Ref Expression
dfin2

Proof of Theorem dfin2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 3112 . . . . . 6
2 eldif 3485 . . . . . 6
31, 2mpbiran 918 . . . . 5
43con2bii 332 . . . 4
54anbi2i 694 . . 3
6 eldif 3485 . . 3
75, 6bitr4i 252 . 2
87ineqri 3691 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  /\wa 369  =wceq 1395  e.wcel 1818   cvv 3109  \cdif 3472  i^icin 3474 This theorem is referenced by:  dfun3  3735  dfin3  3736  invdif  3738  difundi  3749  difindi  3751  difdif2  3754 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-dif 3478  df-in 3482
 Copyright terms: Public domain W3C validator