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Theorem ralrp 10372
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 10356 . . . 4  |-  ( x  e.  RR+  <->  ( x  e.  RR  /\  0  < 
x ) )
21imbi1i 315 . . 3  |-  ( ( x  e.  RR+  ->  ph )  <->  ( ( x  e.  RR  /\  0  <  x )  ->  ph )
)
3 impexp 433 . . 3  |-  ( ( ( x  e.  RR  /\  0  <  x )  ->  ph )  <->  ( x  e.  RR  ->  ( 0  <  x  ->  ph )
) )
42, 3bitri 240 . 2  |-  ( ( x  e.  RR+  ->  ph )  <->  ( x  e.  RR  ->  ( 0  <  x  ->  ph )
) )
54ralbii2 2571 1  |-  ( A. x  e.  RR+  ph  <->  A. x  e.  RR  ( 0  < 
x  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   A.wral 2543   class class class wbr 4023   RRcr 8736   0cc0 8737    < clt 8867   RR+crp 10354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-rp 10355
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