Theorem rexrp 9798

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 9777	. . . 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 2517	1 |- ( E. x e. RR+ ph <-> E. x e. RR ( 0 < x /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2176   E.wrex 2485   class class class wbr 4044   RRcr 7924   0cc0 7925   < clt 8107   RR+crp 9775

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-rab 2493  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-rp 9776

This theorem is referenced by: (None)