Theorem rexrp 11853

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 11834	. . . 4 |- ( x e. RR+ <-> ( x e. RR /\ 0 < x ) )
2	1	anbi1i 731	. . 3 |- ( ( x e. RR+ /\ ph ) <-> ( ( x e. RR /\ 0 < x ) /\ ph ) )
3		anass 681	. . 3 |- ( ( ( x e. RR /\ 0 < x ) /\ ph ) <-> ( x e. RR /\ ( 0 < x /\ ph ) ) )
4	2, 3	bitri 264	. 2 |- ( ( x e. RR+ /\ ph ) <-> ( x e. RR /\ ( 0 < x /\ ph ) ) )
5	4	rexbii2 3039	1 |- ( E. x e. RR+ ph <-> E. x e. RR ( 0 < x /\ ph ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   E.wrex 2913   class class class wbr 4653   RRcr 9935   0cc0 9936   < clt 10074   RR+crp 11832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-rp 11833

This theorem is referenced by: (None)