Theorem ralrp 10055

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 10035	. . . 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 2560	1 |- ( A. x e. RR+ ph <-> A. x e. RR ( 0 < x -> ph ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2209   A.wral 2528   class class class wbr 4125   RRcr 8168   0cc0 8169   < clt 8350   RR+crp 10033

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rab 2537  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-rp 10034

This theorem is referenced by:  caucvgre  11725