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Theorem ssralv 3234
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 3164 . . 3 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21imim1d 75 . 2 (𝐴𝐵 → ((𝑥𝐵𝜑) → (𝑥𝐴𝜑)))
32ralimdv2 2560 1 (𝐴𝐵 → (∀𝑥𝐵 𝜑 → ∀𝑥𝐴 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2160  wral 2468  wss 3144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-ral 2473  df-in 3150  df-ss 3157
This theorem is referenced by:  iinss1  3913  poss  4316  sess2  4356  trssord  4398  funco  5275  funimaexglem  5318  isores3  5837  isoini2  5841  smores  6318  smores2  6320  tfrlem5  6340  resixp  6760  ac6sfi  6927  difinfinf  7131  peano5nnnn  7922  peano5nni  8953  caucvgre  11025  rexanuz  11032  cau3lem  11158  isumclim3  11466  fsumiun  11520  pcfac  12385  ctinf  12484  strsetsid  12548  imasaddfnlemg  12794  tgcn  14185  tgcnp  14186  cnss2  14204  cncnp  14207  sslm  14224  metrest  14483  rescncf  14545  suplociccex  14580  limcresi  14612  nninfsellemeq  15242
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