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Theorem ralrp 9048
Description: Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
Assertion
Ref Expression
ralrp  |-  ( A. x  e.  RR+  ph  <->  A. x  e.  RR  ( 0  < 
x  ->  ph ) )

Proof of Theorem ralrp
StepHypRef Expression
1 elrp 9029 . . . 4  |-  ( x  e.  RR+  <->  ( x  e.  RR  /\  0  < 
x ) )
21imbi1i 236 . . 3  |-  ( ( x  e.  RR+  ->  ph )  <->  ( ( x  e.  RR  /\  0  <  x )  ->  ph )
)
3 impexp 259 . . 3  |-  ( ( ( x  e.  RR  /\  0  <  x )  ->  ph )  <->  ( x  e.  RR  ->  ( 0  <  x  ->  ph )
) )
42, 3bitri 182 . 2  |-  ( ( x  e.  RR+  ->  ph )  <->  ( x  e.  RR  ->  ( 0  <  x  ->  ph )
) )
54ralbii2 2382 1  |-  ( A. x  e.  RR+  ph  <->  A. x  e.  RR  ( 0  < 
x  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1434   A.wral 2353   class class class wbr 3811   RRcr 7250   0cc0 7251    < clt 7423   RR+crp 9027
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rab 2362  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-rp 9028
This theorem is referenced by:  caucvgre  10239
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