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Theorem eceq2 8740
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 6045 . 2 (𝐴 = 𝐵 → (𝐴 “ {𝐶}) = (𝐵 “ {𝐶}))
2 df-ec 8702 . 2 [𝐶]𝐴 = (𝐴 “ {𝐶})
3 df-ec 8702 . 2 [𝐶]𝐵 = (𝐵 “ {𝐶})
41, 2, 33eqtr4g 2789 1 (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  {csn 4621  cima 5670  [cec 8698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ec 8702
This theorem is referenced by:  eceq2i  8741  eceq2d  8742  qseq2  8755  qusval  17489  efgrelexlemb  19662  efgcpbllemb  19667  znzrh2  21410
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