Theorem ssdifssd 3405

Description: If A is contained in B, then (A \ C) is also contained in B. Deduction form of ssdifss 3398. (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 3398	. 2 |- (A C_ B -> (A \ C) C_ B)
3	1, 2	syl 15	1 |- (ph -> (A \ C) C_ B)

Syntax hints:   -> wi 4   \ cdif 3207   C_ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260

This theorem is referenced by: (None)