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

Theorem ssralv 3004
 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 2939 . . 3 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21imim1d 69 . 2 (𝐴𝐵 → ((𝑥𝐵𝜑) → (𝑥𝐴𝜑)))
32ralimdv2 2389 1 (𝐴𝐵 → (∀𝑥𝐵 𝜑 → ∀𝑥𝐴 𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1393  ∀wral 2306   ⊆ wss 2917 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-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-in 2924  df-ss 2931 This theorem is referenced by:  iinss1  3669  poss  4035  sess2  4075  trssord  4117  funco  4940  funimaexglem  4982  isores3  5455  isoini2  5458  smores  5907  smores2  5909  tfrlem5  5930  ac6sfi  6352  peano5nnnn  6966  peano5nni  7917  caucvgre  9580  rexanuz  9587  cau3lem  9710
 Copyright terms: Public domain W3C validator