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Theorem ssralv 3074
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 3008 . . 3 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21imim1d 74 . 2 (𝐴𝐵 → ((𝑥𝐵𝜑) → (𝑥𝐴𝜑)))
32ralimdv2 2439 1 (𝐴𝐵 → (∀𝑥𝐵 𝜑 → ∀𝑥𝐴 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1436  wral 2355  wss 2988
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-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-ral 2360  df-in 2994  df-ss 3001
This theorem is referenced by:  iinss1  3725  poss  4098  sess2  4138  trssord  4180  funco  5016  funimaexglem  5059  isores3  5549  isoini2  5553  smores  6005  smores2  6007  tfrlem5  6027  ac6sfi  6560  peano5nnnn  7364  peano5nni  8353  caucvgre  10302  rexanuz  10309  cau3lem  10435  nninfsellemeq  11336
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