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Theorem ssdif2d 3076
 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 3074 . 2 (φ → (A𝐷) ⊆ (A𝐶))
3 ssdifd.1 . . 3 (φAB)
43ssdifd 3073 . 2 (φ → (A𝐶) ⊆ (B𝐶))
52, 4sstrd 2949 1 (φ → (A𝐷) ⊆ (B𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∖ cdif 2908   ⊆ wss 2911 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925 This theorem is referenced by: (None)
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