Theorem eldifd 3211

Description: If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3210. (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 3210	. 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 3198

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203

This theorem is referenced by:  exmidundif  4302  exmidundifim  4303  frirrg  4453  dcdifsnid  6715  phpelm  7096  findcard2d  7123  findcard2sd  7124  diffifi  7126  unsnfidcex  7155  unsnfidcel  7156  undifdcss  7158  difinfsnlem  7358  difinfsn  7359  hashunlem  11130  seq3coll  11169  fsum3cvg  12019  isumss  12032  fisumss  12033  fproddccvg  12213  fprodssdc  12231  sqrt2irr0  12816  nnoddn2prmb  12915  bassetsnn  13219  logbgcd1irr  15778  2lgslem2  15911