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Theorem dfif3 3868
 Description: Alternate definition of the conditional operator df-if 3855. 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 3857 . 2
2 dfif3.1 . . . . . 6
3 biidd 240 . . . . . . 7
43cbvabv 2552 . . . . . 6
52, 4eqtri 2450 . . . . 5
65ineq2i 3604 . . . 4
7 dfrab3 3691 . . . 4
86, 7eqtr4i 2453 . . 3
9 dfrab3 3691 . . . 4
10 notab 3686 . . . . . 6
115difeq2i 3523 . . . . . 6
1210, 11eqtr4i 2453 . . . . 5
1312ineq2i 3604 . . . 4
149, 13eqtr2i 2451 . . 3
158, 14uneq12i 3561 . 2
161, 15eqtr4i 2453 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wceq 1437  cab 2414  crab 2718  cvv 3022   cdif 3376   cun 3377   cin 3378  cif 3854 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ral 2719  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-if 3855 This theorem is referenced by:  dfif4  3869
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