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Theorem dfif3 3371
 Description: Alternate definition of the conditional operator df-if 3360. 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 3361 . 2
2 dfif3.1 . . . . . 6
3 biidd 165 . . . . . . 7
43cbvabv 2177 . . . . . 6
52, 4eqtri 2076 . . . . 5
65ineq2i 3163 . . . 4
7 dfrab3 3241 . . . 4
86, 7eqtr4i 2079 . . 3
9 dfrab3 3241 . . . 4
10 notab 3235 . . . . . 6
115difeq2i 3087 . . . . . 6
1210, 11eqtr4i 2079 . . . . 5
1312ineq2i 3163 . . . 4
149, 13eqtr2i 2077 . . 3
158, 14uneq12i 3123 . 2
161, 15eqtr4i 2079 1
 Colors of variables: wff set class Syntax hints:   wn 3   wceq 1259  cab 2042  crab 2327  cvv 2574   cdif 2942   cun 2943   cin 2944  cif 3359 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-if 3360 This theorem is referenced by: (None)
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