Theorem eceq2 6657

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 5017	. 2 |- ( A = B -> ( A " { C } ) = ( B " { C } ) )
2		df-ec 6622	. 2 |- [ C ] A = ( A " { C } )
3		df-ec 6622	. 2 |- [ C ] B = ( B " { C } )
4	1, 2, 3	3eqtr4g 2263	1 |- ( A = B -> [ C ] A = [ C ] B )

Syntax hints:   -> wi 4   = wceq 1373   {csn 3633   "cima 4678   [cec 6618

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-cnv 4683  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-ec 6622

This theorem is referenced by:  eceq2i  6658  eceq2d  6659  qseq2  6671  nqnq0pi  7551  qusval  13155  qusex  13157  znzrh2  14408