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Theorem ssralv 3234
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 3164 . . 3  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21imim1d 75 . 2  |-  ( A 
C_  B  ->  (
( x  e.  B  ->  ph )  ->  (
x  e.  A  ->  ph ) ) )
32ralimdv2 2560 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 2160   A.wral 2468    C_ 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  6317  smores2  6319  tfrlem5  6339  resixp  6759  ac6sfi  6926  difinfinf  7130  peano5nnnn  7921  peano5nni  8952  caucvgre  11022  rexanuz  11029  cau3lem  11155  isumclim3  11463  fsumiun  11517  pcfac  12382  ctinf  12481  strsetsid  12545  imasaddfnlemg  12791  tgcn  14165  tgcnp  14166  cnss2  14184  cncnp  14187  sslm  14204  metrest  14463  rescncf  14525  suplociccex  14560  limcresi  14592  nninfsellemeq  15222
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