Theorem eceq2 6629

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 5004	. 2 |- ( A = B -> ( A " { C } ) = ( B " { C } ) )
2		df-ec 6594	. 2 |- [ C ] A = ( A " { C } )
3		df-ec 6594	. 2 |- [ C ] B = ( B " { C } )
4	1, 2, 3	3eqtr4g 2254	1 |- ( A = B -> [ C ] A = [ C ] B )

Syntax hints:   -> wi 4   = wceq 1364   {csn 3622   "cima 4666   [cec 6590

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-cnv 4671  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-ec 6594

This theorem is referenced by:  eceq2i  6630  eceq2d  6631  qseq2  6643  nqnq0pi  7505  qusval  12966  qusex  12968  znzrh2  14202