Theorem difeq1 3467

Description: Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
difeq1	|- ( A = B -> ( A \ C ) = ( B \ C ) )

Proof of Theorem difeq1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabeq 2966	. 2 |- ( A = B -> { x e. A | -. x e. C } = { x e. B | -. x e. C } )
2		dfdif2 3337	. 2 |- ( A \ C ) = { x e. A | -. x e. C }
3		dfdif2 3337	. 2 |- ( B \ C ) = { x e. B | -. x e. C }
4	1, 2, 3	3eqtr4g 2500	1 |- ( A = B -> ( A \ C ) = ( B \ C ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1369   e. wcel 1756   {crab 2719   \ cdif 3325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-rab 2724  df-dif 3331

This theorem is referenced by:  difeq12  3469  difeq1i  3470  difeq1d  3473  uneqdifeq  3767  hartogslem1  7756  kmlem9  8327  kmlem11  8329  kmlem12  8330  isfin1a  8461  fin1a2lem13  8581  incexclem  13299  ismri  14569  f1otrspeq  15953  pmtrval  15957  pmtrfrn  15964  symgsssg  15973  symgfisg  15974  symggen  15976  psgnunilem1  15999  psgnunilem5  16000  psgneldm  16009  ablfac1eulem  16573  islbs  17157  lbsextlem1  17239  lbsextlem2  17240  lbsextlem3  17241  lbsextlem4  17242  zrhcofipsgn  18023  submafval  18390  lpval  18743  bwthOLD  19014  2ndcdisj  19060  isufil  19476  ptcmplem2  19625  mblsplit  21015  voliunlem3  21033  ig1pval  21644  nb3graprlem2  23360  iscusgra  23364  isuvtx  23396  difeq  25899  sigaval  26553  issiga  26554  issgon  26566  probun  26802  ballotlemgval  26906  cvmscbv  27147  cvmsi  27154  cvmsval  27155  symdifeq1  27851  sdrgacs  29558  compne  29696  isfrgra  30582  1vwmgra  30595  3vfriswmgra  30597  m1detdiag  30934  ldepsnlinc  31050