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Theorem dfif3 3742
 Description: Alternate definition of the conditional operator df-if 3733. Note that is independent of i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro, 8-Sep-2013.)
Hypothesis
Ref Expression
dfif3.1
Assertion
Ref Expression
dfif3
Distinct variable group:   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem dfif3
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfif6 3735 . 2
2 dfif3.1 . . . . . 6
3 biidd 229 . . . . . . 7
43cbvabv 2555 . . . . . 6
52, 4eqtri 2456 . . . . 5
65ineq2i 3532 . . . 4
7 dfrab3 3610 . . . 4
86, 7eqtr4i 2459 . . 3
9 dfrab3 3610 . . . 4
10 notab 3604 . . . . . 6
115difeq2i 3455 . . . . . 6
1210, 11eqtr4i 2459 . . . . 5
1312ineq2i 3532 . . . 4
149, 13eqtr2i 2457 . . 3
158, 14uneq12i 3492 . 2
161, 15eqtr4i 2459 1
 Colors of variables: wff set class Syntax hints:   wn 3   wceq 1652  cab 2422  crab 2702  cvv 2949   cdif 3310   cun 3311   cin 3312  cif 3732 This theorem is referenced by:  dfif4  3743 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2703  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-if 3733
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