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Theorem ralrp 11311
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 11294 . . . 4  |-  ( x  e.  RR+  <->  ( x  e.  RR  /\  0  < 
x ) )
21imbi1i 331 . . 3  |-  ( ( x  e.  RR+  ->  ph )  <->  ( ( x  e.  RR  /\  0  <  x )  ->  ph )
)
3 impexp 452 . . 3  |-  ( ( ( x  e.  RR  /\  0  <  x )  ->  ph )  <->  ( x  e.  RR  ->  ( 0  <  x  ->  ph )
) )
42, 3bitri 257 . 2  |-  ( ( x  e.  RR+  ->  ph )  <->  ( x  e.  RR  ->  ( 0  <  x  ->  ph )
) )
54ralbii2 2802 1  |-  ( A. x  e.  RR+  ph  <->  A. x  e.  RR  ( 0  < 
x  ->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    e. wcel 1891   A.wral 2737   class class class wbr 4374   RRcr 9525   0cc0 9526    < clt 9662   RR+crp 11292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ral 2742  df-rab 2746  df-v 3015  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-if 3850  df-sn 3937  df-pr 3939  df-op 3943  df-br 4375  df-rp 11293
This theorem is referenced by: (None)
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