Theorem ralrp 7246

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 7233	. . . 4 |- (x e. RR+ <-> (x e. RR /\ 0 < x))
2	1	imbi1i 203	. . 3 |- ((x e. RR+ -> ph) <-> ((x e. RR /\ 0 < x) -> ph))
3		impexp 374	. . 3 |- (((x e. RR /\ 0 < x) -> ph) <-> (x e. RR -> (0 < x -> ph)))
4	2, 3	bitri 190	. 2 |- ((x e. RR+ -> ph) <-> (x e. RR -> (0 < x -> ph)))
5	4	ralbii2 2131	1 |- (A.x e. RR+ ph <-> A.x e. RR (0 < x -> ph))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   e. wcel 1300  A.wral 2105   class class class wbr 3338  RRcr 6385  0cc0 6386  RR+crp 6453   < clt 6653

This theorem is referenced by:  clm4fi 8342  clmnnsi 8344  clmfnn 8353  iscau5 9219  lmbrnns 9220  iscaunns 9222

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rab 2112  df-v 2294  df-un 2600  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-rp 7232

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