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Theorem dfif5 3957
Description: Alternate definition of the conditional operator df-if 3942. Note that is independent of i.e. a constant true or false (see also abvor0 3803). (Contributed by GĂ©rard Lang, 18-Aug-2013.)
Hypothesis
Ref Expression
dfif3.1
Assertion
Ref Expression
dfif5
Distinct variable group:   ,

Proof of Theorem dfif5
StepHypRef Expression
1 inindi 3714 . 2
2 dfif3.1 . . 3
32dfif4 3956 . 2
4 undir 3746 . . 3
5 unidm 3646 . . . . . . . 8
65uneq1i 3653 . . . . . . 7
7 unass 3660 . . . . . . 7
8 undi 3744 . . . . . . 7
96, 7, 83eqtr3ri 2495 . . . . . 6
10 undi 3744 . . . . . . . 8
11 undifabs 3905 . . . . . . . . 9
1211ineq1i 3695 . . . . . . . 8
13 inabs 3728 . . . . . . . 8
1410, 12, 133eqtri 2490 . . . . . . 7
15 undif2 3904 . . . . . . . . 9
1615ineq1i 3695 . . . . . . . 8
17 undi 3744 . . . . . . . 8
1816, 17, 83eqtr4i 2496 . . . . . . 7
1914, 18uneq12i 3655 . . . . . 6
209, 19eqtr4i 2489 . . . . 5
21 unundi 3664 . . . . 5
2220, 21eqtr4i 2489 . . . 4
23 unass 3660 . . . . . 6
24 undi 3744 . . . . . . . . 9
25 uncom 3647 . . . . . . . . 9
26 undif2 3904 . . . . . . . . . 10
2726ineq1i 3695 . . . . . . . . 9
2824, 25, 273eqtr4i 2496 . . . . . . . 8
29 undi 3744 . . . . . . . 8
3028, 29eqtr4i 2489 . . . . . . 7
31 undi 3744 . . . . . . . 8
32 undifabs 3905 . . . . . . . . 9
3332ineq1i 3695 . . . . . . . 8
34 inabs 3728 . . . . . . . 8
3531, 33, 343eqtrri 2491 . . . . . . 7
3630, 35uneq12i 3655 . . . . . 6
37 unidm 3646 . . . . . . 7
3837uneq2i 3654 . . . . . 6
3923, 36, 383eqtr3ri 2495 . . . . 5
40 uncom 3647 . . . . . . 7
4140ineq2i 3696 . . . . . 6
42 undir 3746 . . . . . 6
4341, 42eqtr4i 2489 . . . . 5
44 unundi 3664 . . . . 5
4539, 43, 443eqtr4i 2496 . . . 4
4622, 45ineq12i 3697 . . 3
474, 46eqtr4i 2489 . 2
481, 3, 473eqtr4i 2496 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 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
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