Theorem eceq2 6782

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 5077	. 2 |- ( A = B -> ( A " { C } ) = ( B " { C } ) )
2		df-ec 6747	. 2 |- [ C ] A = ( A " { C } )
3		df-ec 6747	. 2 |- [ C ] B = ( B " { C } )
4	1, 2, 3	3eqtr4g 2289	1 |- ( A = B -> [ C ] A = [ C ] B )

Syntax hints:   -> wi 4   = wceq 1398   {csn 3673   "cima 4734   [cec 6743

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-ec 6747

This theorem is referenced by:  eceq2i  6783  eceq2d  6784  qseq2  6796  nqnq0pi  7718  qusval  13486  qusex  13488  znzrh2  14742