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Theorem rexrp 11322
Description: Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)
Assertion
Ref Expression
rexrp  |-  ( E. x  e.  RR+  ph  <->  E. x  e.  RR  ( 0  < 
x  /\  ph ) )

Proof of Theorem rexrp
StepHypRef Expression
1 elrp 11304 . . . 4  |-  ( x  e.  RR+  <->  ( x  e.  RR  /\  0  < 
x ) )
21anbi1i 701 . . 3  |-  ( ( x  e.  RR+  /\  ph ) 
<->  ( ( x  e.  RR  /\  0  < 
x )  /\  ph ) )
3 anass 655 . . 3  |-  ( ( ( x  e.  RR  /\  0  <  x )  /\  ph )  <->  ( x  e.  RR  /\  ( 0  <  x  /\  ph ) ) )
42, 3bitri 253 . 2  |-  ( ( x  e.  RR+  /\  ph ) 
<->  ( x  e.  RR  /\  ( 0  <  x  /\  ph ) ) )
54rexbii2 2887 1  |-  ( E. x  e.  RR+  ph  <->  E. x  e.  RR  ( 0  < 
x  /\  ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    e. wcel 1887   E.wrex 2738   class class class wbr 4402   RRcr 9538   0cc0 9539    < clt 9675   RR+crp 11302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-rp 11303
This theorem is referenced by: (None)
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