Theorem ssdif2d 3348

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 3346	. 2 |- ( ph -> ( A \ D ) C_ ( A \ C ) )
3		ssdifd.1	. . 3 |- ( ph -> A C_ B )
4	3	ssdifd 3345	. 2 |- ( ph -> ( A \ C ) C_ ( B \ C ) )
5	2, 4	sstrd 3238	1 |- ( ph -> ( A \ D ) C_ ( B \ C ) )

Syntax hints:   -> wi 4   \ cdif 3198   C_ wss 3201

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  df-in 3207  df-ss 3214

This theorem is referenced by: (None)