Theorem rexuz 11738

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

Assertion

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

Distinct variable group:   n, M

Allowed substitution hint:   ph( n)

Proof of Theorem rexuz

Step	Hyp	Ref	Expression
1		eluz1 11691	. . . 4 |- ( M e. ZZ -> ( n e. ( ZZ>= ` M ) <-> ( n e. ZZ /\ M <_ n ) ) )
2	1	anbi1d 741	. . 3 |- ( M e. ZZ -> ( ( n e. ( ZZ>= ` M ) /\ ph ) <-> ( ( n e. ZZ /\ M <_ n ) /\ ph ) ) )
3		anass 681	. . 3 |- ( ( ( n e. ZZ /\ M <_ n ) /\ ph ) <-> ( n e. ZZ /\ ( M <_ n /\ ph ) ) )
4	2, 3	syl6bb 276	. 2 |- ( M e. ZZ -> ( ( n e. ( ZZ>= ` M ) /\ ph ) <-> ( n e. ZZ /\ ( M <_ n /\ ph ) ) ) )
5	4	rexbidv2 3048	1 |- ( M e. ZZ -> ( E. n e. ( ZZ>= ` M ) ph <-> E. n e. ZZ ( M <_ n /\ ph ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   E.wrex 2913   class class class wbr 4653   ` cfv 5888   <_ cle 10075   ZZcz 11377   ZZ>=cuz 11687

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  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-neg 10269  df-z 11378  df-uz 11688

This theorem is referenced by: (None)