Theorem ssdif2d 3406

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	⊢ (φ → A ⊆ B)
ssdif2d.2	⊢ (φ → C ⊆ D)

Assertion

Ref	Expression
ssdif2d	⊢ (φ → (A ∖ D) ⊆ (B ∖ C))

Proof of Theorem ssdif2d

Step	Hyp	Ref	Expression
1		ssdif2d.2	. . 3 ⊢ (φ → C ⊆ D)
2	1	sscond 3404	. 2 ⊢ (φ → (A ∖ D) ⊆ (A ∖ C))
3		ssdifd.1	. . 3 ⊢ (φ → A ⊆ B)
4	3	ssdifd 3403	. 2 ⊢ (φ → (A ∖ C) ⊆ (B ∖ C))
5	2, 4	sstrd 3283	1 ⊢ (φ → (A ∖ D) ⊆ (B ∖ C))

Syntax hints:   → wi 4   ∖ cdif 3207   ⊆ 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)