Theorem eceq2 6717

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 5063	. 2 |- ( A = B -> ( A " { C } ) = ( B " { C } ) )
2		df-ec 6682	. 2 |- [ C ] A = ( A " { C } )
3		df-ec 6682	. 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 3666   "cima 4722   [cec 6678

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 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-ec 6682

This theorem is referenced by:  eceq2i  6718  eceq2d  6719  qseq2  6731  nqnq0pi  7625  qusval  13356  qusex  13358  znzrh2  14610