Theorem eldifd 3125

Description: If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3124. (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 3124	. 2 |- ( A e. ( B \ C ) <-> ( A e. B /\ -. A e. C ) )
4	1, 2, 3	sylanbrc 414	1 |- ( ph -> A e. ( B \ C ) )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2136   \ cdif 3112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-v 2727  df-dif 3117

This theorem is referenced by:  exmidundif  4184  exmidundifim  4185  frirrg  4327  dcdifsnid  6468  phpelm  6828  findcard2d  6853  findcard2sd  6854  diffifi  6856  unsnfidcex  6881  unsnfidcel  6882  undifdcss  6884  difinfsnlem  7060  difinfsn  7061  hashunlem  10713  seq3coll  10751  fsum3cvg  11315  isumss  11328  fisumss  11329  fproddccvg  11509  fprodssdc  11527  sqrt2irr0  12092  nnoddn2prmb  12190  logbgcd1irr  13485