Theorem ssdifd 3263

Description: If A is contained in B, then ( A \ C ) is contained in ( B \ C ). Deduction form of ssdif 3262. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
ssdifd.1	|- ( ph -> A C_ B )

Assertion

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

Proof of Theorem ssdifd

Step	Hyp	Ref	Expression
1		ssdifd.1	. 2 |- ( ph -> A C_ B )
2		ssdif 3262	. 2 |- ( A C_ B -> ( A \ C ) C_ ( B \ C ) )
3	1, 2	syl 14	1 |- ( ph -> ( A \ C ) C_ ( B \ C ) )

Syntax hints:   -> wi 4   \ cdif 3118   C_ wss 3121

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134

This theorem is referenced by:  ssdif2d  3266  phplem4dom  6840  fisseneq  6909