Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfif5 Structured version   Unicode version

Theorem dfif5 3744
 Description: Alternate definition of the conditional operator df-if 3733. Note that is independent of i.e. a constant true or false (see also abvor0 3638). (Contributed by Gérard Lang, 18-Aug-2013.)
Hypothesis
Ref Expression
dfif3.1
Assertion
Ref Expression
dfif5
Distinct variable group:   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem dfif5
StepHypRef Expression
1 inindi 3551 . 2
2 dfif3.1 . . 3
32dfif4 3743 . 2
4 undir 3583 . . 3
5 unidm 3483 . . . . . . . 8
65uneq1i 3490 . . . . . . 7
7 unass 3497 . . . . . . 7
8 undi 3581 . . . . . . 7
96, 7, 83eqtr3ri 2465 . . . . . 6
10 undi 3581 . . . . . . . 8
11 undifabs 3698 . . . . . . . . 9
1211ineq1i 3531 . . . . . . . 8
13 inabs 3565 . . . . . . . 8
1410, 12, 133eqtri 2460 . . . . . . 7
15 undif2 3697 . . . . . . . . 9
1615ineq1i 3531 . . . . . . . 8
17 undi 3581 . . . . . . . 8
1816, 17, 83eqtr4i 2466 . . . . . . 7
1914, 18uneq12i 3492 . . . . . 6
209, 19eqtr4i 2459 . . . . 5
21 unundi 3501 . . . . 5
2220, 21eqtr4i 2459 . . . 4
23 unass 3497 . . . . . 6
24 undi 3581 . . . . . . . . 9
25 uncom 3484 . . . . . . . . 9
26 undif2 3697 . . . . . . . . . 10
2726ineq1i 3531 . . . . . . . . 9
2824, 25, 273eqtr4i 2466 . . . . . . . 8
29 undi 3581 . . . . . . . 8
3028, 29eqtr4i 2459 . . . . . . 7
31 undi 3581 . . . . . . . 8
32 undifabs 3698 . . . . . . . . 9
3332ineq1i 3531 . . . . . . . 8
34 inabs 3565 . . . . . . . 8
3531, 33, 343eqtrri 2461 . . . . . . 7
3630, 35uneq12i 3492 . . . . . 6
37 unidm 3483 . . . . . . 7
3837uneq2i 3491 . . . . . 6
3923, 36, 383eqtr3ri 2465 . . . . 5
40 uncom 3484 . . . . . . 7
4140ineq2i 3532 . . . . . 6
42 undir 3583 . . . . . 6
4341, 42eqtr4i 2459 . . . . 5
44 unundi 3501 . . . . 5
4539, 43, 443eqtr4i 2466 . . . 4
4622, 45ineq12i 3533 . . 3
474, 46eqtr4i 2459 . 2
481, 3, 473eqtr4i 2466 1
 Colors of variables: wff set class Syntax hints:   wceq 1652  cab 2422  cvv 2949   cdif 3310   cun 3311   cin 3312  cif 3732 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733
 Copyright terms: Public domain W3C validator