Theorem ralrp 6290

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 6283	. . . 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 1674	1 |- (A.x e. RR+ ph <-> A.x e. RR (0 < x -> ph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 960  A.wral 1648   class class class wbr 2624  RRcr 5245  0cc0 5246  RR+crp 5312   < clt 5498

This theorem is referenced by:  clm4f 7082  clmnns 7084  clmfnn 7093  iscau5 7938  lmbrnns 7939  iscaunns 7941

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rab 1655  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-rp 6282

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