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Theorem ralrp 10586
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 10570 . . . 4  |-  ( x  e.  RR+  <->  ( x  e.  RR  /\  0  < 
x ) )
21imbi1i 316 . . 3  |-  ( ( x  e.  RR+  ->  ph )  <->  ( ( x  e.  RR  /\  0  <  x )  ->  ph )
)
3 impexp 434 . . 3  |-  ( ( ( x  e.  RR  /\  0  <  x )  ->  ph )  <->  ( x  e.  RR  ->  ( 0  <  x  ->  ph )
) )
42, 3bitri 241 . 2  |-  ( ( x  e.  RR+  ->  ph )  <->  ( x  e.  RR  ->  ( 0  <  x  ->  ph )
) )
54ralbii2 2694 1  |-  ( A. x  e.  RR+  ph  <->  A. x  e.  RR  ( 0  < 
x  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1721   A.wral 2666   class class class wbr 4172   RRcr 8945   0cc0 8946    < clt 9076   RR+crp 10568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-rp 10569
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