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Theorem eceq2 8786
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 6073 . 2 (𝐴 = 𝐵 → (𝐴 “ {𝐶}) = (𝐵 “ {𝐶}))
2 df-ec 8747 . 2 [𝐶]𝐴 = (𝐴 “ {𝐶})
3 df-ec 8747 . 2 [𝐶]𝐵 = (𝐵 “ {𝐶})
41, 2, 33eqtr4g 2802 1 (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  {csn 4626  cima 5688  [cec 8743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747
This theorem is referenced by:  eceq2i  8787  eceq2d  8788  qseq2  8802  qusval  17587  efgrelexlemb  19768  efgcpbllemb  19773  znzrh2  21564
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