Theorem eldifd 3207

Description: If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3206. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
eldifd.1	|- ( ph -> A e. B )
eldifd.2	|- ( ph -> -. A e. C )

Assertion

Ref	Expression
eldifd	|- ( ph -> A e. ( B \ C ) )

Proof of Theorem eldifd

Step	Hyp	Ref	Expression
1		eldifd.1	. 2 |- ( ph -> A e. B )
2		eldifd.2	. 2 |- ( ph -> -. A e. C )
3		eldif 3206	. 2 |- ( A e. ( B \ C ) <-> ( A e. B /\ -. A e. C ) )
4	1, 2, 3	sylanbrc 417	1 |- ( ph -> A e. ( B \ C ) )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2200   \ cdif 3194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199

This theorem is referenced by:  exmidundif  4291  exmidundifim  4292  frirrg  4442  dcdifsnid  6663  phpelm  7041  findcard2d  7066  findcard2sd  7067  diffifi  7069  unsnfidcex  7098  unsnfidcel  7099  undifdcss  7101  difinfsnlem  7282  difinfsn  7283  hashunlem  11043  seq3coll  11082  fsum3cvg  11910  isumss  11923  fisumss  11924  fproddccvg  12104  fprodssdc  12122  sqrt2irr0  12707  nnoddn2prmb  12806  bassetsnn  13110  logbgcd1irr  15662  2lgslem2  15792