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Theorem ssdif2d 4118
 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 4116 . 2 (𝜑 → (𝐴𝐷) ⊆ (𝐴𝐶))
3 ssdifd.1 . . 3 (𝜑𝐴𝐵)
43ssdifd 4115 . 2 (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐶))
52, 4sstrd 3975 1 (𝜑 → (𝐴𝐷) ⊆ (𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∖ cdif 3931   ⊆ wss 3934 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-dif 3937  df-in 3941  df-ss 3950 This theorem is referenced by:  mblfinlem3  34918  mblfinlem4  34919
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