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Theorem dfif4 3924
 Description: Alternate definition of the conditional operator df-if 3910. 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 3923 . 2
3 undir 3722 . 2
4 undi 3720 . . . 4
5 undi 3720 . . . . 5
6 uncom 3610 . . . . . 6
7 unvdif 3869 . . . . . 6
86, 7ineq12i 3662 . . . . 5
9 inv1 3789 . . . . 5
105, 8, 93eqtri 2455 . . . 4
114, 10ineq12i 3662 . . 3
12 inass 3672 . . 3
1311, 12eqtri 2451 . 2
142, 3, 133eqtri 2455 1
 Colors of variables: wff setvar class Syntax hints:   wceq 1437  cab 2407  cvv 3081   cdif 3433   cun 3434   cin 3435  cif 3909 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910 This theorem is referenced by:  dfif5  3925
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