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Theorem eceq2 6472
Description: Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
Assertion
Ref Expression
eceq2 (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)

Proof of Theorem eceq2
StepHypRef Expression
1 imaeq1 4882 . 2 (𝐴 = 𝐵 → (𝐴 “ {𝐶}) = (𝐵 “ {𝐶}))
2 df-ec 6437 . 2 [𝐶]𝐴 = (𝐴 “ {𝐶})
3 df-ec 6437 . 2 [𝐶]𝐵 = (𝐵 “ {𝐶})
41, 2, 33eqtr4g 2198 1 (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1332  {csn 3530  cima 4548  [cec 6433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3078  df-in 3080  df-ss 3087  df-sn 3536  df-pr 3537  df-op 3539  df-br 3936  df-opab 3996  df-cnv 4553  df-dm 4555  df-rn 4556  df-res 4557  df-ima 4558  df-ec 6437
This theorem is referenced by:  qseq2  6484  nqnq0pi  7268
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