Theorem difeq1 4071

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 3413	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐶} = {𝑥 ∈ 𝐵 ∣ ¬ 𝑥 ∈ 𝐶})
2		dfdif2 3910	. 2 ⊢ (𝐴 ∖ 𝐶) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐶}
3		dfdif2 3910	. 2 ⊢ (𝐵 ∖ 𝐶) = {𝑥 ∈ 𝐵 ∣ ¬ 𝑥 ∈ 𝐶}
4	1, 2, 3	3eqtr4g 2796	1 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2113  {crab 3399   ∖ cdif 3898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-dif 3904

This theorem is referenced by:  difeq12  4073  difeq1i  4074  difeq1d  4077  symdifeq1  4207  uneqdifeq  4445  hartogslem1  9447  kmlem9  10069  kmlem11  10071  kmlem12  10072  isfin1a  10202  fin1a2lem13  10322  fundmge2nop0  14425  incexclem  15759  coprmprod  16588  coprmproddvds  16590  ismri  17554  f1otrspeq  19376  pmtrval  19380  pmtrfrn  19387  symgsssg  19396  symgfisg  19397  symggen  19399  psgnunilem1  19422  psgnunilem5  19423  psgneldm  19432  ablfac1eulem  20003  sdrgacs  20734  islbs  21028  lbsextlem1  21113  lbsextlem2  21114  lbsextlem3  21115  lbsextlem4  21116  cofipsgn  21548  selvffval  22076  submafval  22523  m1detdiag  22541  lpval  23083  2ndcdisj  23400  isufil  23847  ptcmplem2  23997  mblsplit  25489  voliunlem3  25509  ig1pval  26137  nbgr2vtx1edg  29423  nbuhgr2vtx1edgb  29425  nb3grprlem2  29454  uvtx01vtx  29470  cplgr1v  29503  dfconngr1  30263  isconngr1  30265  isfrgr  30335  frgr1v  30346  nfrgr2v  30347  frgr3v  30350  1vwmgr  30351  3vfriswmgr  30353  difeq  32593  symgcntz  33167  tocycval  33190  extvval  33696  sigaval  34268  issiga  34269  issgon  34280  isros  34325  unelros  34328  difelros  34329  inelsros  34335  diffiunisros  34336  rossros  34337  inelcarsg  34468  carsgclctunlem2  34476  probun  34576  ballotlemgval  34681  cvmscbv  35452  cvmsi  35459  cvmsval  35460  poimirlem4  37825  dssmapfvd  44258  compne  44681  dvmptfprod  46189  caragensplit  46744  vonvolmbllem  46904  vonvolmbl  46905  ldepsnlinc  48754  eenglngeehlnm  48985