Theorem eldifd 3210

Description: If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3209. (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 3209	. 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 2202   \ cdif 3197

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202

This theorem is referenced by:  exmidundif  4296  exmidundifim  4297  frirrg  4447  dcdifsnid  6671  phpelm  7052  findcard2d  7079  findcard2sd  7080  diffifi  7082  unsnfidcex  7111  unsnfidcel  7112  undifdcss  7114  difinfsnlem  7297  difinfsn  7298  hashunlem  11066  seq3coll  11105  fsum3cvg  11938  isumss  11951  fisumss  11952  fproddccvg  12132  fprodssdc  12150  sqrt2irr0  12735  nnoddn2prmb  12834  bassetsnn  13138  logbgcd1irr  15690  2lgslem2  15820