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Theorem rexrp 9663
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 9642 . . . 4  |-  ( x  e.  RR+  <->  ( x  e.  RR  /\  0  < 
x ) )
21anbi1i 458 . . 3  |-  ( ( x  e.  RR+  /\  ph ) 
<->  ( ( x  e.  RR  /\  0  < 
x )  /\  ph ) )
3 anass 401 . . 3  |-  ( ( ( x  e.  RR  /\  0  <  x )  /\  ph )  <->  ( x  e.  RR  /\  ( 0  <  x  /\  ph ) ) )
42, 3bitri 184 . 2  |-  ( ( x  e.  RR+  /\  ph ) 
<->  ( x  e.  RR  /\  ( 0  <  x  /\  ph ) ) )
54rexbii2 2488 1  |-  ( E. x  e.  RR+  ph  <->  E. x  e.  RR  ( 0  < 
x  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    e. wcel 2148   E.wrex 2456   class class class wbr 4000   RRcr 7801   0cc0 7802    < clt 7982   RR+crp 9640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-rab 2464  df-v 2739  df-un 3133  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-rp 9641
This theorem is referenced by: (None)
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