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Theorem ssralv 3059
Description: Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)
Assertion
Ref Expression
ssralv  |-  ( A 
C_  B  ->  ( A. x  e.  B  ph 
->  A. x  e.  A  ph ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem ssralv
StepHypRef Expression
1 ssel 2994 . . 3  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21imim1d 74 . 2  |-  ( A 
C_  B  ->  (
( x  e.  B  ->  ph )  ->  (
x  e.  A  ->  ph ) ) )
32ralimdv2 2432 1  |-  ( A 
C_  B  ->  ( A. x  e.  B  ph 
->  A. x  e.  A  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1434   A.wral 2349    C_ wss 2974
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-ral 2354  df-in 2980  df-ss 2987
This theorem is referenced by:  iinss1  3698  poss  4061  sess2  4101  trssord  4143  funco  4970  funimaexglem  5013  isores3  5486  isoini2  5489  smores  5941  smores2  5943  tfrlem5  5963  ac6sfi  6431  peano5nnnn  7120  peano5nni  8109  caucvgre  10005  rexanuz  10012  cau3lem  10138
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