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Theorem ssdif2d 3215
Description: If 𝐴 is contained in 𝐵 and 𝐶 is contained in 𝐷, then (𝐴𝐷) is contained in (𝐵𝐶). 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 3213 . 2 (𝜑 → (𝐴𝐷) ⊆ (𝐴𝐶))
3 ssdifd.1 . . 3 (𝜑𝐴𝐵)
43ssdifd 3212 . 2 (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐶))
52, 4sstrd 3107 1 (𝜑 → (𝐴𝐷) ⊆ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  cdif 3068  wss 3071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084
This theorem is referenced by: (None)
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