Theorem ralrp 8604

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 8585	. . . 4 |- ( x e. RR+ <-> ( x e. RR /\ 0 < x ) )
2	1	imbi1i 227	. . 3 |- ( ( x e. RR+ -> ph ) <-> ( ( x e. RR /\ 0 < x ) -> ph ) )
3		impexp 250	. . 3 |- ( ( ( x e. RR /\ 0 < x ) -> ph ) <-> ( x e. RR -> ( 0 < x -> ph ) ) )
4	2, 3	bitri 173	. 2 |- ( ( x e. RR+ -> ph ) <-> ( x e. RR -> ( 0 < x -> ph ) ) )
5	4	ralbii2 2334	1 |- ( A. x e. RR+ ph <-> A. x e. RR ( 0 < x -> ph ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   e. wcel 1393   A.wral 2306   class class class wbr 3764   RRcr 6888   0cc0 6889   < clt 7060   RR+crp 8583

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-rp 8584

This theorem is referenced by:  caucvgre  9580