Theorem eceq2 6844

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 5121	. 2 |- ( A = B -> ( A " { C } ) = ( B " { C } ) )
2		df-ec 6809	. 2 |- [ C ] A = ( A " { C } )
3		df-ec 6809	. 2 |- [ C ] B = ( B " { C } )
4	1, 2, 3	3eqtr4g 2296	1 |- ( A = B -> [ C ] A = [ C ] B )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   {csn 3709   "cima 4777   [cec 6805

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-cnv 4782  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-ec 6809

This theorem is used by:  eceq2i  6845  eceq2d  6846  qseq2  6858  nqnq0pi  7805  qusval  13644  qusex  13646  znzrh2  14981