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Theorem r19.29uz 12146
Description: A version of 19.29 1606 for upper integer quantifiers. (Contributed by Mario Carneiro, 10-Feb-2014.)
Hypothesis
Ref Expression
rexuz3.1  |-  Z  =  ( ZZ>= `  M )
Assertion
Ref Expression
r19.29uz  |-  ( ( A. k  e.  Z  ph 
/\  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ps )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( ph  /\  ps )
)
Distinct variable groups:    j, M    ph, j    j, k, Z
Allowed substitution hints:    ph( k)    ps( j, k)    M( k)

Proof of Theorem r19.29uz
StepHypRef Expression
1 rexuz3.1 . . . . . . . . 9  |-  Z  =  ( ZZ>= `  M )
21uztrn2 10495 . . . . . . . 8  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
32ex 424 . . . . . . 7  |-  ( j  e.  Z  ->  (
k  e.  ( ZZ>= `  j )  ->  k  e.  Z ) )
4 pm3.2 435 . . . . . . . 8  |-  ( ph  ->  ( ps  ->  ( ph  /\  ps ) ) )
54a1i 11 . . . . . . 7  |-  ( j  e.  Z  ->  ( ph  ->  ( ps  ->  (
ph  /\  ps )
) ) )
63, 5imim12d 70 . . . . . 6  |-  ( j  e.  Z  ->  (
( k  e.  Z  ->  ph )  ->  (
k  e.  ( ZZ>= `  j )  ->  ( ps  ->  ( ph  /\  ps ) ) ) ) )
76ralimdv2 2778 . . . . 5  |-  ( j  e.  Z  ->  ( A. k  e.  Z  ph 
->  A. k  e.  (
ZZ>= `  j ) ( ps  ->  ( ph  /\ 
ps ) ) ) )
87impcom 420 . . . 4  |-  ( ( A. k  e.  Z  ph 
/\  j  e.  Z
)  ->  A. k  e.  ( ZZ>= `  j )
( ps  ->  ( ph  /\  ps ) ) )
9 ralim 2769 . . . 4  |-  ( A. k  e.  ( ZZ>= `  j ) ( ps 
->  ( ph  /\  ps ) )  ->  ( A. k  e.  ( ZZ>=
`  j ) ps 
->  A. k  e.  (
ZZ>= `  j ) (
ph  /\  ps )
) )
108, 9syl 16 . . 3  |-  ( ( A. k  e.  Z  ph 
/\  j  e.  Z
)  ->  ( A. k  e.  ( ZZ>= `  j ) ps  ->  A. k  e.  ( ZZ>= `  j ) ( ph  /\ 
ps ) ) )
1110reximdva 2810 . 2  |-  ( A. k  e.  Z  ph  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ps 
->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) (
ph  /\  ps )
) )
1211imp 419 1  |-  ( ( A. k  e.  Z  ph 
/\  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ps )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( ph  /\  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   ` cfv 5446   ZZ>=cuz 10480
This theorem is referenced by:  caubnd  12154  caucvgb  12465  cvgcmp  12587  ulmcau  20303
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-pre-lttri 9056  ax-pre-lttrn 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-neg 9286  df-z 10275  df-uz 10481
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