Theorem ralrp 9456

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

Step	Hyp	Ref	Expression
1		elrp 9436	. . . 4 |- ( x e. RR+ <-> ( x e. RR /\ 0 < x ) )
2	1	imbi1i 237	. . 3 |- ( ( x e. RR+ -> ph ) <-> ( ( x e. RR /\ 0 < x ) -> ph ) )
3		impexp 261	. . 3 |- ( ( ( x e. RR /\ 0 < x ) -> ph ) <-> ( x e. RR -> ( 0 < x -> ph ) ) )
4	2, 3	bitri 183	. 2 |- ( ( x e. RR+ -> ph ) <-> ( x e. RR -> ( 0 < x -> ph ) ) )
5	4	ralbii2 2443	1 |- ( A. x e. RR+ ph <-> A. x e. RR ( 0 < x -> ph ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 1480   A.wral 2414   class class class wbr 3924   RRcr 7612   0cc0 7613   < clt 7793   RR+crp 9434

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 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rab 2423  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-rp 9435

This theorem is referenced by:  caucvgre  10746