Theorem ssdif2d 3320

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 3318	. 2 ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ (𝐴 ∖ 𝐶))
3		ssdifd.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
4	3	ssdifd 3317	. 2 ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))
5	2, 4	sstrd 3211	1 ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ (𝐵 ∖ 𝐶))

Syntax hints:   → wi 4   ∖ cdif 3171   ⊆ wss 3174

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176  df-in 3180  df-ss 3187

This theorem is referenced by: (None)