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Theorem dfopif 3981
 Description: Rewrite df-op 3823 using . When both arguments are sets, it reduces to the standard Kuratowski definition; otherwise, it is defined to be the empty set. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
dfopif

Proof of Theorem dfopif
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-op 3823 . 2
2 df-3an 938 . . 3
32abbii 2548 . 2
4 iftrue 3745 . . . 4
5 ibar 491 . . . . 5
65abbi2dv 2551 . . . 4
74, 6eqtr2d 2469 . . 3
8 pm2.21 102 . . . . . . 7
98adantrd 455 . . . . . 6
109abssdv 3417 . . . . 5
11 ss0 3658 . . . . 5
1210, 11syl 16 . . . 4
13 iffalse 3746 . . . 4
1412, 13eqtr4d 2471 . . 3
157, 14pm2.61i 158 . 2
161, 3, 153eqtri 2460 1
 Colors of variables: wff set class Syntax hints:   wn 3   wa 359   w3a 936   wceq 1652   wcel 1725  cab 2422  cvv 2956   wss 3320  c0 3628  cif 3739  csn 3814  cpr 3815  cop 3817 This theorem is referenced by:  dfopg  3982  opeq1  3984  opeq2  3985  nfop  4000  opprc  4005  opex  4427 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-3an 938  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-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-op 3823
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