Theorem eceq2 6734

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 5069	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 “ {𝐶}) = (𝐵 “ {𝐶}))
2		df-ec 6699	. 2 ⊢ [𝐶]𝐴 = (𝐴 “ {𝐶})
3		df-ec 6699	. 2 ⊢ [𝐶]𝐵 = (𝐵 “ {𝐶})
4	1, 2, 3	3eqtr4g 2287	1 ⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)

Syntax hints:   → wi 4   = wceq 1395  {csn 3667   “ cima 4726  [cec 6695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-cnv 4731  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-ec 6699

This theorem is referenced by:  eceq2i  6735  eceq2d  6736  qseq2  6748  nqnq0pi  7648  qusval  13396  qusex  13398  znzrh2  14650