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Theorem dfif4 3673
 Description: Alternate definition of the conditional operator df-if 3663. 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:   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem dfif4
StepHypRef Expression
1 dfif3.1 . . 3
21dfif3 3672 . 2
3 undir 3504 . 2
4 undi 3502 . . . 4
5 undi 3502 . . . . 5
6 uncom 3408 . . . . . 6
7 undifv 3624 . . . . . 6
86, 7ineq12i 3455 . . . . 5
9 inv1 3577 . . . . 5
105, 8, 93eqtri 2377 . . . 4
114, 10ineq12i 3455 . . 3
12 inass 3465 . . 3
1311, 12eqtri 2373 . 2
142, 3, 133eqtri 2377 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:  dfif5  3674
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