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Theorem dfif3 3672
 Description: Alternate definition of the conditional operator df-if 3663. 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 3665 . 2
2 dfif3.1 . . . . . 6
3 biidd 228 . . . . . . 7
43cbvabv 2472 . . . . . 6
52, 4eqtri 2373 . . . . 5
65ineq2i 3454 . . . 4
7 dfrab3 3531 . . . 4
86, 7eqtr4i 2376 . . 3
9 dfrab3 3531 . . . 4
10 notab 3525 . . . . . 6
115difeq2i 3382 . . . . . 6
1210, 11eqtr4i 2376 . . . . 5
1312ineq2i 3454 . . . 4
149, 13eqtr2i 2374 . . 3
158, 14uneq12i 3416 . 2
161, 15eqtr4i 2376 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wceq 1642  cab 2339  crab 2618  cvv 2859   cdif 3206   cun 3207   cin 3208  cif 3662 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-if 3663 This theorem is referenced by:  dfif4  3673
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