Theorem ralrp 9048

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 9029	. . . 4 |- ( x e. RR+ <-> ( x e. RR /\ 0 < x ) )
2	1	imbi1i 236	. . 3 |- ( ( x e. RR+ -> ph ) <-> ( ( x e. RR /\ 0 < x ) -> ph ) )
3		impexp 259	. . 3 |- ( ( ( x e. RR /\ 0 < x ) -> ph ) <-> ( x e. RR -> ( 0 < x -> ph ) ) )
4	2, 3	bitri 182	. 2 |- ( ( x e. RR+ -> ph ) <-> ( x e. RR -> ( 0 < x -> ph ) ) )
5	4	ralbii2 2382	1 |- ( A. x e. RR+ ph <-> A. x e. RR ( 0 < x -> ph ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   e. wcel 1434   A.wral 2353   class class class wbr 3811   RRcr 7250   0cc0 7251   < clt 7423   RR+crp 9027

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-ral 2358  df-rab 2362  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-rp 9028

This theorem is referenced by:  caucvgre  10239