Theorem eldifd 3086

Description: If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3085. (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 3085	. 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 1481   \ cdif 3073

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078

This theorem is referenced by:  exmidundif  4137  exmidundifim  4138  frirrg  4280  dcdifsnid  6408  phpelm  6768  findcard2d  6793  findcard2sd  6794  diffifi  6796  unsnfidcex  6816  unsnfidcel  6817  undifdcss  6819  difinfsnlem  6992  difinfsn  6993  hashunlem  10582  seq3coll  10617  fsum3cvg  11179  isumss  11192  fisumss  11193  fproddccvg  11373  sqrt2irr0  11878  logbgcd1irr  13092