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Theorem dfif4 3752
 Description: Alternate definition of the conditional operator df-if 3742. 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 3751 . 2
3 undir 3592 . 2
4 undi 3590 . . . 4
5 undi 3590 . . . . 5
6 uncom 3493 . . . . . 6
7 undifv 3704 . . . . . 6
86, 7ineq12i 3542 . . . . 5
9 inv1 3656 . . . . 5
105, 8, 93eqtri 2462 . . . 4
114, 10ineq12i 3542 . . 3
12 inass 3553 . . 3
1311, 12eqtri 2458 . 2
142, 3, 133eqtri 2462 1
 Colors of variables: wff set class Syntax hints:   wceq 1653  cab 2424  cvv 2958   cdif 3319   cun 3320   cin 3321  cif 3741 This theorem is referenced by:  dfif5  3753 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742
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