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Theorem r19.29uz 12081
Description: A version of 19.29 1603 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 10435 . . . . . . . 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 2729 . . . . 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 2720 . . . 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 2761 . 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 1649    e. wcel 1717   A.wral 2649   E.wrex 2650   ` cfv 5394   ZZ>=cuz 10420
This theorem is referenced by:  caubnd  12089  caucvgb  12400  cvgcmp  12522  ulmcau  20178
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-pre-lttri 8997  ax-pre-lttrn 8998
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-neg 9226  df-z 10215  df-uz 10421
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