Theorem eldifd 3081

Description: If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3080. (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 3080	. 2 |- ( A e. ( B \ C ) <-> ( A e. B /\ -. A e. C ) )
4	1, 2, 3	sylanbrc 413	1 |- ( ph -> A e. ( B \ C ) )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 1480   \ cdif 3068

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073

This theorem is referenced by:  exmidundif  4129  exmidundifim  4130  frirrg  4272  dcdifsnid  6400  phpelm  6760  findcard2d  6785  findcard2sd  6786  diffifi  6788  unsnfidcex  6808  unsnfidcel  6809  undifdcss  6811  difinfsnlem  6984  difinfsn  6985  hashunlem  10550  seq3coll  10585  fsum3cvg  11147  isumss  11160  fisumss  11161  fproddccvg  11341  sqrt2irr0  11842