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Theorem ssdif2d 3711
 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 (𝜑𝐴𝐵)
ssdif2d.2 (𝜑𝐶𝐷)
Assertion
Ref Expression
ssdif2d (𝜑 → (𝐴𝐷) ⊆ (𝐵𝐶))

Proof of Theorem ssdif2d
StepHypRef Expression
1 ssdif2d.2 . . 3 (𝜑𝐶𝐷)
21sscond 3709 . 2 (𝜑 → (𝐴𝐷) ⊆ (𝐴𝐶))
3 ssdifd.1 . . 3 (𝜑𝐴𝐵)
43ssdifd 3708 . 2 (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐶))
52, 4sstrd 3578 1 (𝜑 → (𝐴𝐷) ⊆ (𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∖ cdif 3537   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554 This theorem is referenced by:  mblfinlem3  32618  mblfinlem4  32619
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