Theorem eceq2 6166

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

Step	Hyp	Ref	Expression
1		imaeq1 4683	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 “ {𝐶}) = (𝐵 “ {𝐶}))
2		df-ec 6131	. 2 ⊢ [𝐶]𝐴 = (𝐴 “ {𝐶})
3		df-ec 6131	. 2 ⊢ [𝐶]𝐵 = (𝐵 “ {𝐶})
4	1, 2, 3	3eqtr4g 2138	1 ⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)

Syntax hints:   → wi 4   = wceq 1284  {csn 3398   “ cima 4366  [cec 6127

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-ec 6131

This theorem is referenced by:  qseq2  6178  nqnq0pi  6628