Theorem ssdif2d 4102

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 4100	. 2 ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ (𝐴 ∖ 𝐶))
3		ssdifd.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
4	3	ssdifd 4099	. 2 ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))
5	2, 4	sstrd 3948	1 ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ (𝐵 ∖ 𝐶))

This proof depends on syntax axioms:   → wi 4   ∖ cdif 3903   ⊆ wss 3906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-dif 3909  df-ss 3923

This theorem is used by:  elrgspnsubrunlem2  33634  mblfinlem3  38369  mblfinlem4  38370