Theorem eceq2 6680

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 5036	. 2 |- ( A = B -> ( A " { C } ) = ( B " { C } ) )
2		df-ec 6645	. 2 |- [ C ] A = ( A " { C } )
3		df-ec 6645	. 2 |- [ C ] B = ( B " { C } )
4	1, 2, 3	3eqtr4g 2265	1 |- ( A = B -> [ C ] A = [ C ] B )

Syntax hints:   -> wi 4   = wceq 1373   {csn 3643   "cima 4696   [cec 6641

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-ec 6645

This theorem is referenced by:  eceq2i  6681  eceq2d  6682  qseq2  6694  nqnq0pi  7586  qusval  13270  qusex  13272  znzrh2  14523