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Theorem dfif5 3888
 Description: Alternate definition of the conditional operator df-if 3873. Note that is independent of i.e. a constant true or false (see also ab0orv 3753). (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 3640 . 2
2 dfif3.1 . . 3
32dfif4 3887 . 2
4 undir 3683 . . 3
5 unidm 3568 . . . . . . . 8
65uneq1i 3575 . . . . . . 7
7 unass 3582 . . . . . . 7
8 undi 3681 . . . . . . 7
96, 7, 83eqtr3ri 2502 . . . . . 6
10 undi 3681 . . . . . . . 8
11 undifabs 3835 . . . . . . . . 9
1211ineq1i 3621 . . . . . . . 8
13 inabs 3665 . . . . . . . 8
1410, 12, 133eqtri 2497 . . . . . . 7
15 undif2 3834 . . . . . . . . 9
1615ineq1i 3621 . . . . . . . 8
17 undi 3681 . . . . . . . 8
1816, 17, 83eqtr4i 2503 . . . . . . 7
1914, 18uneq12i 3577 . . . . . 6
209, 19eqtr4i 2496 . . . . 5
21 unundi 3586 . . . . 5
2220, 21eqtr4i 2496 . . . 4
23 unass 3582 . . . . . 6
24 undi 3681 . . . . . . . . 9
25 uncom 3569 . . . . . . . . 9
26 undif2 3834 . . . . . . . . . 10
2726ineq1i 3621 . . . . . . . . 9
2824, 25, 273eqtr4i 2503 . . . . . . . 8
29 undi 3681 . . . . . . . 8
3028, 29eqtr4i 2496 . . . . . . 7
31 undi 3681 . . . . . . . 8
32 undifabs 3835 . . . . . . . . 9
3332ineq1i 3621 . . . . . . . 8
34 inabs 3665 . . . . . . . 8
3531, 33, 343eqtrri 2498 . . . . . . 7
3630, 35uneq12i 3577 . . . . . 6
37 unidm 3568 . . . . . . 7
3837uneq2i 3576 . . . . . 6
3923, 36, 383eqtr3ri 2502 . . . . 5
40 uncom 3569 . . . . . . 7
4140ineq2i 3622 . . . . . 6
42 undir 3683 . . . . . 6
4341, 42eqtr4i 2496 . . . . 5
44 unundi 3586 . . . . 5
4539, 43, 443eqtr4i 2503 . . . 4
4622, 45ineq12i 3623 . . 3
474, 46eqtr4i 2496 . 2
481, 3, 473eqtr4i 2503 1
 Colors of variables: wff setvar class Syntax hints:   wceq 1452  cab 2457  cvv 3031   cdif 3387   cun 3388   cin 3389  cif 3872 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873 This theorem is referenced by: (None)
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