Theorem rexrp 9910

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 9889	. . . 4 |- ( x e. RR+ <-> ( x e. RR /\ 0 < x ) )
2	1	anbi1i 458	. . 3 |- ( ( x e. RR+ /\ ph ) <-> ( ( x e. RR /\ 0 < x ) /\ ph ) )
3		anass 401	. . 3 |- ( ( ( x e. RR /\ 0 < x ) /\ ph ) <-> ( x e. RR /\ ( 0 < x /\ ph ) ) )
4	2, 3	bitri 184	. 2 |- ( ( x e. RR+ /\ ph ) <-> ( x e. RR /\ ( 0 < x /\ ph ) ) )
5	4	rexbii2 2543	1 |- ( E. x e. RR+ ph <-> E. x e. RR ( 0 < x /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2202   E.wrex 2511   class class class wbr 4088   RRcr 8030   0cc0 8031   < clt 8213   RR+crp 9887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-rab 2519  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-rp 9888

This theorem is referenced by: (None)