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Theorem dfopif 4122
 Description: Rewrite df-op 3943 using . When both arguments are sets, it reduces to the standard Kuratowski definition; otherwise, it is defined to be the empty set. Avoid directly depending on this detail so that theorems will not depend on the Kuratowski construction. (Contributed by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)
Assertion
Ref Expression
dfopif

Proof of Theorem dfopif
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-op 3943 . 2
2 df-3an 984 . . 3
32abbii 2539 . 2
4 iftrue 3855 . . . 4
5 ibar 506 . . . . 5
65abbi2dv 2542 . . . 4
74, 6eqtr2d 2458 . . 3
8 pm2.21 111 . . . . . . 7
98adantrd 469 . . . . . 6
109abssdv 3473 . . . . 5
11 ss0 3733 . . . . 5
1210, 11syl 17 . . . 4
13 iffalse 3858 . . . 4
1412, 13eqtr4d 2460 . . 3
157, 14pm2.61i 167 . 2
161, 3, 153eqtri 2449 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wa 370   w3a 982   wceq 1437   wcel 1872  cab 2409  cvv 3017   wss 3374  c0 3699  cif 3849  csn 3936  cpr 3938  cop 3942 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 2058  ax-ext 2403 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-v 3019  df-dif 3377  df-in 3381  df-ss 3388  df-nul 3700  df-if 3850  df-op 3943 This theorem is referenced by:  dfopg  4123  opeq1  4125  opeq2  4126  nfop  4141  opprc  4147  opex  4623  csbopg2  31632
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