ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ssdifd GIF version

Theorem ssdifd 3120
Description: If 𝐴 is contained in 𝐵, then (𝐴𝐶) is contained in (𝐵𝐶). Deduction form of ssdif 3119. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
ssdifd.1 (𝜑𝐴𝐵)
Assertion
Ref Expression
ssdifd (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐶))

Proof of Theorem ssdifd
StepHypRef Expression
1 ssdifd.1 . 2 (𝜑𝐴𝐵)
2 ssdif 3119 . 2 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
31, 2syl 14 1 (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  cdif 2981  wss 2984
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-dif 2986  df-in 2990  df-ss 2997
This theorem is referenced by:  ssdif2d  3123  phplem4dom  6508  fisseneq  6567
  Copyright terms: Public domain W3C validator