Theorem eceq2 7784

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 5461	. 2 |- ( A = B -> ( A " { C } ) = ( B " { C } ) )
2		df-ec 7744	. 2 |- [ C ] A = ( A " { C } )
3		df-ec 7744	. 2 |- [ C ] B = ( B " { C } )
4	1, 2, 3	3eqtr4g 2681	1 |- ( A = B -> [ C ] A = [ C ] B )

Syntax hints:   -> wi 4   = wceq 1483   {csn 4177   "cima 5117   [cec 7740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744

This theorem is referenced by:  qseq2  7797  qusval  16202  efgrelexlemb  18163  efgcpbllemb  18168  vrgpfval  18179  znzrh2  19894  eceq2i  34040  eceq2d  34041