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Theorem cokeq1 4230
 Description: Equality theorem for Kuratowski composition of two classes. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
cokeq1 (A = B → (A k C) = (B k C))

Proof of Theorem cokeq1
StepHypRef Expression
1 ins2keq 4218 . . . 4 (A = BIns2k A = Ins2k B)
21ineq1d 3456 . . 3 (A = B → ( Ins2k AIns3k kC) = ( Ins2k BIns3k kC))
32imakeq1d 4228 . 2 (A = B → (( Ins2k AIns3k kC) “k V) = (( Ins2k BIns3k kC) “k V))
4 df-cok 4190 . 2 (A k C) = (( Ins2k AIns3k kC) “k V)
5 df-cok 4190 . 2 (B k C) = (( Ins2k BIns3k kC) “k V)
63, 4, 53eqtr4g 2410 1 (A = B → (A k C) = (B k C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642  Vcvv 2859   ∩ cin 3208  ◡kccnvk 4175   Ins2k cins2k 4176   Ins3k cins3k 4177   “k cimak 4179   ∘k ccomk 4180 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-3an 936  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-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ins2k 4187  df-imak 4189  df-cok 4190 This theorem is referenced by:  cokeq1i  4232  cokeq1d  4234
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