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Theorem dfif5 3674
 Description: Alternate definition of the conditional operator df-if 3663. Note that is independent of i.e. a constant true or false (see also abvor0 3567). (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 3472 . 2
2 dfif3.1 . . 3
32dfif4 3673 . 2
4 undir 3504 . . 3
5 unidm 3407 . . . . . . . 8
65uneq1i 3414 . . . . . . 7
7 unass 3420 . . . . . . 7
8 undi 3502 . . . . . . 7
96, 7, 83eqtr3ri 2382 . . . . . 6
10 undi 3502 . . . . . . . 8
11 undifabs 3627 . . . . . . . . . 10
1211ineq1i 3453 . . . . . . . . 9
13 inabs 3486 . . . . . . . . 9
1412, 13eqtri 2373 . . . . . . . 8
1510, 14eqtri 2373 . . . . . . 7
16 undif2 3626 . . . . . . . . 9
1716ineq1i 3453 . . . . . . . 8
18 undi 3502 . . . . . . . 8
1917, 18, 83eqtr4i 2383 . . . . . . 7
2015, 19uneq12i 3416 . . . . . 6
219, 20eqtr4i 2376 . . . . 5
22 unundi 3424 . . . . 5
2321, 22eqtr4i 2376 . . . 4
24 unass 3420 . . . . . 6
25 undi 3502 . . . . . . . . 9
26 uncom 3408 . . . . . . . . 9
27 undif2 3626 . . . . . . . . . 10
2827ineq1i 3453 . . . . . . . . 9
2925, 26, 283eqtr4i 2383 . . . . . . . 8
30 undi 3502 . . . . . . . 8
3129, 30eqtr4i 2376 . . . . . . 7
32 undi 3502 . . . . . . . 8
33 undifabs 3627 . . . . . . . . 9
3433ineq1i 3453 . . . . . . . 8
35 inabs 3486 . . . . . . . 8
3632, 34, 353eqtrri 2378 . . . . . . 7
3731, 36uneq12i 3416 . . . . . 6
38 unidm 3407 . . . . . . 7
3938uneq2i 3415 . . . . . 6
4024, 37, 393eqtr3ri 2382 . . . . 5
41 uncom 3408 . . . . . . 7
4241ineq2i 3454 . . . . . 6
43 undir 3504 . . . . . 6
4442, 43eqtr4i 2376 . . . . 5
45 unundi 3424 . . . . 5
4640, 44, 453eqtr4i 2383 . . . 4
4723, 46ineq12i 3455 . . 3
484, 47eqtr4i 2376 . 2
491, 3, 483eqtr4i 2383 1
 Colors of variables: wff setvar class Syntax hints:   wceq 1642  cab 2339  cvv 2859   cdif 3206   cun 3207   cin 3208  cif 3662 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-if 3663 This theorem is referenced by: (None)
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