Theorem eceq1 5963

Description: Equality theorem for equivalence class. (Contributed by set.mm contributors, 23-Jul-1995.)

Assertion

Ref	Expression
eceq1	|- (A = B -> [A]C = [B]C)

Proof of Theorem eceq1

Step	Hyp	Ref	Expression
1		sneq 3745	. . 3 |- (A = B -> {A} = {B})
2	1	imaeq2d 4943	. 2 |- (A = B -> (C"{A}) = (C"{B}))
3		df-ec 5948	. 2 |- [A]C = (C"{A})
4		df-ec 5948	. 2 |- [B]C = (C"{B})
5	2, 3, 4	3eqtr4g 2410	1 |- (A = B -> [A]C = [B]C)

Syntax hints:   -> wi 4   = wceq 1642  {csn 3738  "cima 4723  [cec 5946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-sn 3742  df-ima 4728  df-ec 5948

This theorem is referenced by:  ecelqsg  5980  snec  5988  ecoptocl  5997  nceq  6109