Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfif4 Unicode version

Theorem dfif4 3956
 Description: Alternate definition of the conditional operator df-if 3942. Note that is independent of i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.)
Hypothesis
Ref Expression
dfif3.1
Assertion
Ref Expression
dfif4
Distinct variable group:   ,

Proof of Theorem dfif4
StepHypRef Expression
1 dfif3.1 . . 3
21dfif3 3955 . 2
3 undir 3746 . 2
4 undi 3744 . . . 4
5 undi 3744 . . . . 5
6 uncom 3647 . . . . . 6
7 unvdif 3902 . . . . . 6
86, 7ineq12i 3697 . . . . 5
9 inv1 3812 . . . . 5
105, 8, 93eqtri 2490 . . . 4
114, 10ineq12i 3697 . . 3
12 inass 3707 . . 3
1311, 12eqtri 2486 . 2
142, 3, 133eqtri 2490 1
 Colors of variables: wff setvar class Syntax hints:  =wceq 1395  {cab 2442   cvv 3109  \cdif 3472  u.cun 3473  i^icin 3474  ifcif 3941 This theorem is referenced by:  dfif5  3957 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-or 370  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-ne 2654  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942
 Copyright terms: Public domain W3C validator