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Theorem ralrp 11014
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 10998 . . . 4  |-  ( x  e.  RR+  <->  ( x  e.  RR  /\  0  < 
x ) )
21imbi1i 325 . . 3  |-  ( ( x  e.  RR+  ->  ph )  <->  ( ( x  e.  RR  /\  0  <  x )  ->  ph )
)
3 impexp 446 . . 3  |-  ( ( ( x  e.  RR  /\  0  <  x )  ->  ph )  <->  ( x  e.  RR  ->  ( 0  <  x  ->  ph )
) )
42, 3bitri 249 . 2  |-  ( ( x  e.  RR+  ->  ph )  <->  ( x  e.  RR  ->  ( 0  <  x  ->  ph )
) )
54ralbii2 2748 1  |-  ( A. x  e.  RR+  ph  <->  A. x  e.  RR  ( 0  < 
x  ->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1756   A.wral 2720   class class class wbr 4297   RRcr 9286   0cc0 9287    < clt 9423   RR+crp 10996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298  df-rp 10997
This theorem is referenced by: (None)
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