Theorem ralrp 9750

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 9730	. . . 4 |- ( x e. RR+ <-> ( x e. RR /\ 0 < x ) )
2	1	imbi1i 238	. . 3 |- ( ( x e. RR+ -> ph ) <-> ( ( x e. RR /\ 0 < x ) -> ph ) )
3		impexp 263	. . 3 |- ( ( ( x e. RR /\ 0 < x ) -> ph ) <-> ( x e. RR -> ( 0 < x -> ph ) ) )
4	2, 3	bitri 184	. 2 |- ( ( x e. RR+ -> ph ) <-> ( x e. RR -> ( 0 < x -> ph ) ) )
5	4	ralbii2 2507	1 |- ( A. x e. RR+ ph <-> A. x e. RR ( 0 < x -> ph ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2167   A.wral 2475   class class class wbr 4033   RRcr 7878   0cc0 7879   < clt 8061   RR+crp 9728

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rab 2484  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-rp 9729

This theorem is referenced by:  caucvgre  11146