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Theorem eceq1 5962
 Description: Equality theorem for equivalence class. (Contributed by set.mm contributors, 23-Jul-1995.)
Assertion
Ref Expression
eceq1 (A = B → [A]C = [B]C)

Proof of Theorem eceq1
StepHypRef Expression
1 sneq 3744 . . 3 (A = B → {A} = {B})
21imaeq2d 4942 . 2 (A = B → (C “ {A}) = (C “ {B}))
3 df-ec 5947 . 2 [A]C = (C “ {A})
4 df-ec 5947 . 2 [B]C = (C “ {B})
52, 3, 43eqtr4g 2410 1 (A = B → [A]C = [B]C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642  {csn 3737   “ cima 4722  [cec 5945 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-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-sn 3741  df-ima 4727  df-ec 5947 This theorem is referenced by:  ecelqsg  5979  snec  5987  ecoptocl  5996  nceq  6108
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