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Theorem rexrp 8907
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 8887 . . . 4  |-  ( x  e.  RR+  <->  ( x  e.  RR  /\  0  < 
x ) )
21anbi1i 446 . . 3  |-  ( ( x  e.  RR+  /\  ph ) 
<->  ( ( x  e.  RR  /\  0  < 
x )  /\  ph ) )
3 anass 393 . . 3  |-  ( ( ( x  e.  RR  /\  0  <  x )  /\  ph )  <->  ( x  e.  RR  /\  ( 0  <  x  /\  ph ) ) )
42, 3bitri 182 . 2  |-  ( ( x  e.  RR+  /\  ph ) 
<->  ( x  e.  RR  /\  ( 0  <  x  /\  ph ) ) )
54rexbii2 2382 1  |-  ( E. x  e.  RR+  ph  <->  E. x  e.  RR  ( 0  < 
x  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    e. wcel 1434   E.wrex 2354   class class class wbr 3805   RRcr 7112   0cc0 7113    < clt 7285   RR+crp 8885
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-rab 2362  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-rp 8886
This theorem is referenced by: (None)
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