Theorem ssdif2d 3368

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 3366	. 2 |- ( ph -> ( A \ D ) C_ ( A \ C ) )
3		ssdifd.1	. . 3 |- ( ph -> A C_ B )
4	3	ssdifd 3365	. 2 |- ( ph -> ( A \ C ) C_ ( B \ C ) )
5	2, 4	sstrd 3258	1 |- ( ph -> ( A \ D ) C_ ( B \ C ) )

Syntax hints:   -> wi 4   \ cdif 3217   C_ wss 3220

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  df-in 3226  df-ss 3233

This theorem is referenced by: (None)