Theorem ssdif2d 3749

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 3747	. 2 |- ( ph -> ( A \ D ) C_ ( A \ C ) )
3		ssdifd.1	. . 3 |- ( ph -> A C_ B )
4	3	ssdifd 3746	. 2 |- ( ph -> ( A \ C ) C_ ( B \ C ) )
5	2, 4	sstrd 3613	1 |- ( ph -> ( A \ D ) C_ ( B \ C ) )

Syntax hints:   -> wi 4   \ cdif 3571   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588

This theorem is referenced by:  mblfinlem3  33448  mblfinlem4  33449