Theorem rexuz2 9519

Description: Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)

Assertion

Ref	Expression
rexuz2	|- ( E. n e. ( ZZ>= ` M ) ph <-> ( M e. ZZ /\ E. n e. ZZ ( M <_ n /\ ph ) ) )

Distinct variable group:   n, M

Allowed substitution hint:   ph( n)

Proof of Theorem rexuz2

Step	Hyp	Ref	Expression
1		eluz2 9472	. . . . . 6 |- ( n e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ n e. ZZ /\ M <_ n ) )
2		df-3an 970	. . . . . 6 |- ( ( M e. ZZ /\ n e. ZZ /\ M <_ n ) <-> ( ( M e. ZZ /\ n e. ZZ ) /\ M <_ n ) )
3	1, 2	bitri 183	. . . . 5 |- ( n e. ( ZZ>= ` M ) <-> ( ( M e. ZZ /\ n e. ZZ ) /\ M <_ n ) )
4	3	anbi1i 454	. . . 4 |- ( ( n e. ( ZZ>= ` M ) /\ ph ) <-> ( ( ( M e. ZZ /\ n e. ZZ ) /\ M <_ n ) /\ ph ) )
5		anass 399	. . . . 5 |- ( ( ( ( M e. ZZ /\ n e. ZZ ) /\ M <_ n ) /\ ph ) <-> ( ( M e. ZZ /\ n e. ZZ ) /\ ( M <_ n /\ ph ) ) )
6		anass 399	. . . . . 6 |- ( ( ( M e. ZZ /\ n e. ZZ ) /\ ( M <_ n /\ ph ) ) <-> ( M e. ZZ /\ ( n e. ZZ /\ ( M <_ n /\ ph ) ) ) )
7		an12 551	. . . . . 6 |- ( ( M e. ZZ /\ ( n e. ZZ /\ ( M <_ n /\ ph ) ) ) <-> ( n e. ZZ /\ ( M e. ZZ /\ ( M <_ n /\ ph ) ) ) )
8	6, 7	bitri 183	. . . . 5 |- ( ( ( M e. ZZ /\ n e. ZZ ) /\ ( M <_ n /\ ph ) ) <-> ( n e. ZZ /\ ( M e. ZZ /\ ( M <_ n /\ ph ) ) ) )
9	5, 8	bitri 183	. . . 4 |- ( ( ( ( M e. ZZ /\ n e. ZZ ) /\ M <_ n ) /\ ph ) <-> ( n e. ZZ /\ ( M e. ZZ /\ ( M <_ n /\ ph ) ) ) )
10	4, 9	bitri 183	. . 3 |- ( ( n e. ( ZZ>= ` M ) /\ ph ) <-> ( n e. ZZ /\ ( M e. ZZ /\ ( M <_ n /\ ph ) ) ) )
11	10	rexbii2 2477	. 2 |- ( E. n e. ( ZZ>= ` M ) ph <-> E. n e. ZZ ( M e. ZZ /\ ( M <_ n /\ ph ) ) )
12		r19.42v 2623	. 2 |- ( E. n e. ZZ ( M e. ZZ /\ ( M <_ n /\ ph ) ) <-> ( M e. ZZ /\ E. n e. ZZ ( M <_ n /\ ph ) ) )
13	11, 12	bitri 183	1 |- ( E. n e. ( ZZ>= ` M ) ph <-> ( M e. ZZ /\ E. n e. ZZ ( M <_ n /\ ph ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   /\ w3a 968   e. wcel 2136   E.wrex 2445   class class class wbr 3982   ` cfv 5188   <_ cle 7934   ZZcz 9191   ZZ>=cuz 9466

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-cnex 7844  ax-resscn 7845

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-ov 5845  df-neg 8072  df-z 9192  df-uz 9467

This theorem is referenced by:  2rexuz  9520