Theorem ralrp 6226

Description: Quantification over positive reals.

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 6220	. . . 4 |- (x e. RR+ <-> (x e. RR /\ 0 < x))
2	1	imbi1i 186	. . 3 |- ((x e. RR+ -> ph) <-> ((x e. RR /\ 0 < x) -> ph))
3		impexp 347	. . 3 |- (((x e. RR /\ 0 < x) -> ph) <-> (x e. RR -> (0 < x -> ph)))
4	2, 3	bitr 173	. 2 |- ((x e. RR+ -> ph) <-> (x e. RR -> (0 < x -> ph)))
5	4	ralbii2 1663	1 |- (A.x e. RR+ ph <-> A.x e. RR (0 < x -> ph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 955  A.wral 1637   class class class wbr 2609  RRcr 5205  0cc0 5206  RR+crp 5272   < clt 5458

This theorem is referenced by:  clm4f 7020  clmnns 7022  iscau5 7878

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-rab 1644  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-rp 6219

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