Theorem eceq2 5964

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

Assertion

Ref	Expression
eceq2	⊢ (A = B → [C]A = [C]B)

Proof of Theorem eceq2

Step	Hyp	Ref	Expression
1		imaeq1 4938	. 2 ⊢ (A = B → (A “ {C}) = (B “ {C}))
2		df-ec 5948	. 2 ⊢ [C]A = (A “ {C})
3		df-ec 5948	. 2 ⊢ [C]B = (B “ {C})
4	1, 2, 3	3eqtr4g 2410	1 ⊢ (A = B → [C]A = [C]B)

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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-rex 2621  df-br 4641  df-ima 4728  df-ec 5948

This theorem is referenced by:  qseq2  5976