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

Theorem ssralv 3131
Description: Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)
Assertion
Ref Expression
ssralv (𝐴𝐵 → (∀𝑥𝐵 𝜑 → ∀𝑥𝐴 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ssralv
StepHypRef Expression
1 ssel 3061 . . 3 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21imim1d 75 . 2 (𝐴𝐵 → ((𝑥𝐵𝜑) → (𝑥𝐴𝜑)))
32ralimdv2 2479 1 (𝐴𝐵 → (∀𝑥𝐵 𝜑 → ∀𝑥𝐴 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1465  wral 2393  wss 3041
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-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-ral 2398  df-in 3047  df-ss 3054
This theorem is referenced by:  iinss1  3795  poss  4190  sess2  4230  trssord  4272  funco  5133  funimaexglem  5176  isores3  5684  isoini2  5688  smores  6157  smores2  6159  tfrlem5  6179  resixp  6595  ac6sfi  6760  difinfinf  6954  peano5nnnn  7668  peano5nni  8691  caucvgre  10721  rexanuz  10728  cau3lem  10854  isumclim3  11160  fsumiun  11214  ctinf  11870  strsetsid  11919  tgcn  12304  tgcnp  12305  cnss2  12323  cncnp  12326  sslm  12343  metrest  12602  rescncf  12664  suplociccex  12699  limcresi  12731  nninfsellemeq  13137
  Copyright terms: Public domain W3C validator