Theorem eldifd 3230

Description: If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3229. (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 3229	. 2 |- ( A e. ( B \ C ) <-> ( A e. B /\ -. A e. C ) )
4	1, 2, 3	sylanbrc 421	1 |- ( ph -> A e. ( B \ C ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   e. wcel 2209   \ cdif 3217

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222

This theorem is used by:  exmidundif  4345  exmidundifim  4346  frirrg  4497  dcdifsnid  6781  phpelm  7172  findcard2d  7199  findcard2sd  7200  diffifi  7202  unsnfidcex  7231  unsnfidcel  7232  undifdcss  7234  difinfsnlem  7445  difinfsn  7446  hashunlem  11266  hashf1lem1  11307  seq3coll  11316  fsum3cvg  12170  isumss  12183  fisumss  12184  fproddccvg  12364  fprodssdc  12382  sqrt2irr0  12968  nnoddn2prmb  13070  bassetsnn  13467  gsumresfi  14263  logbgcd1irr  16203  2lgslem2  16416