Theorem ssdif2d 3260

Description: If A is contained in B and C is contained in D, then ( A \ D ) is contained in ( B \ C ). Deduction form. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
ssdifd.1	|- ( ph -> A C_ B )
ssdif2d.2	|- ( ph -> C C_ D )

Assertion

Ref	Expression
ssdif2d	|- ( ph -> ( A \ D ) C_ ( B \ C ) )

Proof of Theorem ssdif2d

Step	Hyp	Ref	Expression
1		ssdif2d.2	. . 3 |- ( ph -> C C_ D )
2	1	sscond 3258	. 2 |- ( ph -> ( A \ D ) C_ ( A \ C ) )
3		ssdifd.1	. . 3 |- ( ph -> A C_ B )
4	3	ssdifd 3257	. 2 |- ( ph -> ( A \ C ) C_ ( B \ C ) )
5	2, 4	sstrd 3151	1 |- ( ph -> ( A \ D ) C_ ( B \ C ) )

Syntax hints:   -> wi 4   \ cdif 3112   C_ wss 3115

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-v 2727  df-dif 3117  df-in 3121  df-ss 3128

This theorem is referenced by: (None)