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Theorem ssralv 3032
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 2967 . . 3 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21imim1d 73 . 2 (𝐴𝐵 → ((𝑥𝐵𝜑) → (𝑥𝐴𝜑)))
32ralimdv2 2406 1 (𝐴𝐵 → (∀𝑥𝐵 𝜑 → ∀𝑥𝐴 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1409  wral 2323  wss 2945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-ral 2328  df-in 2952  df-ss 2959
This theorem is referenced by:  iinss1  3697  poss  4063  sess2  4103  trssord  4145  funco  4968  funimaexglem  5010  isores3  5483  isoini2  5486  smores  5938  smores2  5940  tfrlem5  5961  ac6sfi  6383  peano5nnnn  7024  peano5nni  7993  caucvgre  9808  rexanuz  9815  cau3lem  9941
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