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Theorem rexrp 9486
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 9465 . . . 4  |-  ( x  e.  RR+  <->  ( x  e.  RR  /\  0  < 
x ) )
21anbi1i 453 . . 3  |-  ( ( x  e.  RR+  /\  ph ) 
<->  ( ( x  e.  RR  /\  0  < 
x )  /\  ph ) )
3 anass 398 . . 3  |-  ( ( ( x  e.  RR  /\  0  <  x )  /\  ph )  <->  ( x  e.  RR  /\  ( 0  <  x  /\  ph ) ) )
42, 3bitri 183 . 2  |-  ( ( x  e.  RR+  /\  ph ) 
<->  ( x  e.  RR  /\  ( 0  <  x  /\  ph ) ) )
54rexbii2 2446 1  |-  ( E. x  e.  RR+  ph  <->  E. x  e.  RR  ( 0  < 
x  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    e. wcel 1480   E.wrex 2417   class class class wbr 3932   RRcr 7638   0cc0 7639    < clt 7819   RR+crp 9463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-rab 2425  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3933  df-rp 9464
This theorem is referenced by: (None)
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