Theorem ssdifssd 3285

Description: If A is contained in B, then ( A \ C ) is also contained in B. Deduction form of ssdifss 3277. (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 3277	. 2 |- ( A C_ B -> ( A \ C ) C_ B )
3	1, 2	syl 14	1 |- ( ph -> ( A \ C ) C_ B )

Syntax hints:   -> wi 4   \ cdif 3138   C_ wss 3141

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-dif 3143  df-in 3147  df-ss 3154

This theorem is referenced by:  zfz1isolem1  10833