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 ) )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2209   \ cdif 3217

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 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 theorem 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 referenced by:  exmidundif  4338  exmidundifim  4339  frirrg  4490  dcdifsnid  6767  phpelm  7158  findcard2d  7185  findcard2sd  7186  diffifi  7188  unsnfidcex  7217  unsnfidcel  7218  undifdcss  7220  difinfsnlem  7429  difinfsn  7430  hashunlem  11222  hashf1lem1  11263  seq3coll  11272  fsum3cvg  12123  isumss  12136  fisumss  12137  fproddccvg  12317  fprodssdc  12335  sqrt2irr0  12920  nnoddn2prmb  13019  bassetsnn  13387  logbgcd1irr  15992  2lgslem2  16125