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Theorem eceq2 8749
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 6054 . 2 (𝐴 = 𝐵 → (𝐴 “ {𝐶}) = (𝐵 “ {𝐶}))
2 df-ec 8711 . 2 [𝐶]𝐴 = (𝐴 “ {𝐶})
3 df-ec 8711 . 2 [𝐶]𝐵 = (𝐵 “ {𝐶})
41, 2, 33eqtr4g 2796 1 (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  {csn 4628  cima 5679  [cec 8707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ec 8711
This theorem is referenced by:  eceq2i  8750  eceq2d  8751  qseq2  8764  qusval  17495  efgrelexlemb  19666  efgcpbllemb  19671  znzrh2  21411
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