Theorem ssdifssd 3288

Description: If A is contained in B, then ( A \ C ) is also contained in B. Deduction form of ssdifss 3280. (Contributed by David Moews, 1-May-2017.)

Hypothesis

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

Assertion

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

Proof of Theorem ssdifssd

Step	Hyp	Ref	Expression
1		ssdifd.1	. 2 |- ( ph -> A C_ B )
2		ssdifss 3280	. 2 |- ( A C_ B -> ( A \ C ) C_ B )
3	1, 2	syl 14	1 |- ( ph -> ( A \ C ) C_ B )

Syntax hints:   -> wi 4   \ cdif 3141   C_ wss 3144

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157

This theorem is referenced by:  zfz1isolem1  10838