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

Theorem ssrabdv 3176
Description: Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 31-Aug-2006.)
Hypotheses
Ref Expression
ssrabdv.1 (𝜑𝐵𝐴)
ssrabdv.2 ((𝜑𝑥𝐵) → 𝜓)
Assertion
Ref Expression
ssrabdv (𝜑𝐵 ⊆ {𝑥𝐴𝜓})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem ssrabdv
StepHypRef Expression
1 ssrabdv.1 . 2 (𝜑𝐵𝐴)
2 ssrabdv.2 . . 3 ((𝜑𝑥𝐵) → 𝜓)
32ralrimiva 2505 . 2 (𝜑 → ∀𝑥𝐵 𝜓)
4 ssrab 3175 . 2 (𝐵 ⊆ {𝑥𝐴𝜓} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜓))
51, 3, 4sylanbrc 413 1 (𝜑𝐵 ⊆ {𝑥𝐴𝜓})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1480  wral 2416  {crab 2420  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-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-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rab 2425  df-in 3077  df-ss 3084
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator