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Theorem eceq2 6230
 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 4713 . 2 (𝐴 = 𝐵 → (𝐴 “ {𝐶}) = (𝐵 “ {𝐶}))
2 df-ec 6195 . 2 [𝐶]𝐴 = (𝐴 “ {𝐶})
3 df-ec 6195 . 2 [𝐶]𝐵 = (𝐵 “ {𝐶})
41, 2, 33eqtr4g 2140 1 (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1285  {csn 3416   “ cima 4394  [cec 6191 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-cnv 4399  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-ec 6195 This theorem is referenced by:  qseq2  6242  nqnq0pi  6742
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