Theorem rexrp 9603

Description: Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)

Assertion

Ref	Expression
rexrp	|- ( E. x e. RR+ ph <-> E. x e. RR ( 0 < x /\ ph ) )

Proof of Theorem rexrp

Step	Hyp	Ref	Expression
1		elrp 9582	. . . 4 |- ( x e. RR+ <-> ( x e. RR /\ 0 < x ) )
2	1	anbi1i 454	. . 3 |- ( ( x e. RR+ /\ ph ) <-> ( ( x e. RR /\ 0 < x ) /\ ph ) )
3		anass 399	. . 3 |- ( ( ( x e. RR /\ 0 < x ) /\ ph ) <-> ( x e. RR /\ ( 0 < x /\ ph ) ) )
4	2, 3	bitri 183	. 2 |- ( ( x e. RR+ /\ ph ) <-> ( x e. RR /\ ( 0 < x /\ ph ) ) )
5	4	rexbii2 2475	1 |- ( E. x e. RR+ ph <-> E. x e. RR ( 0 < x /\ ph ) )

Syntax hints:   /\ wa 103   <-> wb 104   e. wcel 2135   E.wrex 2443   class class class wbr 3976   RRcr 7743   0cc0 7744   < clt 7924   RR+crp 9580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rex 2448  df-rab 2451  df-v 2723  df-un 3115  df-sn 3576  df-pr 3577  df-op 3579  df-br 3977  df-rp 9581

This theorem is referenced by: (None)