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Theorem compleq 3243
 Description: Equality law for complement. (Contributed by SF, 11-Jan-2015.)
Assertion
Ref Expression
compleq (A = B → ∼ A = ∼ B)

Proof of Theorem compleq
StepHypRef Expression
1 nineq12 3236 . . 3 ((A = B A = B) → (AA) = (BB))
21anidms 626 . 2 (A = B → (AA) = (BB))
3 df-compl 3212 . 2 A = (AA)
4 df-compl 3212 . 2 B = (BB)
52, 3, 43eqtr4g 2410 1 (A = B → ∼ A = ∼ B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ⩃ cnin 3204   ∼ ccompl 3205 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-v 2861  df-nin 3211  df-compl 3212 This theorem is referenced by:  compleqi  3244  compleqd  3245  difeq2  3247  compleqb  3543  elsuci  4414  nnsucelr  4428  ncfinlower  4483  sfindbl  4530  el2c  6191  nmembers1lem3  6270
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