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Theorem rexfrabdioph 26745
Description: Diophantine set builder for existential quantifier, explicit substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Hypothesis
Ref Expression
rexfrabdioph.1  |-  M  =  ( N  +  1 )
Assertion
Ref Expression
rexfrabdioph  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... M ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. ph }  e.  (Dioph `  M ) )  ->  { u  e.  ( NN0  ^m  ( 1 ... N ) )  |  E. v  e.  NN0  ph }  e.  (Dioph `  N ) )
Distinct variable groups:    u, t,
v, M    t, N, u, v    ph, t
Allowed substitution hints:    ph( v, u)

Proof of Theorem rexfrabdioph
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2540 . . 3  |-  F/_ u
( NN0  ^m  (
1 ... N ) )
2 nfcv 2540 . . 3  |-  F/_ a
( NN0  ^m  (
1 ... N ) )
3 nfv 1626 . . 3  |-  F/ a E. v  e.  NN0  ph
4 nfcv 2540 . . . 4  |-  F/_ u NN0
5 nfsbc1v 3140 . . . 4  |-  F/ u [. a  /  u ]. [. b  /  v ]. ph
64, 5nfrex 2721 . . 3  |-  F/ u E. b  e.  NN0  [. a  /  u ]. [. b  /  v ]. ph
7 nfv 1626 . . . . 5  |-  F/ b
ph
8 nfsbc1v 3140 . . . . 5  |-  F/ v
[. b  /  v ]. ph
9 sbceq1a 3131 . . . . 5  |-  ( v  =  b  ->  ( ph 
<-> 
[. b  /  v ]. ph ) )
107, 8, 9cbvrex 2889 . . . 4  |-  ( E. v  e.  NN0  ph  <->  E. b  e.  NN0  [. b  /  v ]. ph )
11 sbceq1a 3131 . . . . 5  |-  ( u  =  a  ->  ( [. b  /  v ]. ph  <->  [. a  /  u ]. [. b  /  v ]. ph ) )
1211rexbidv 2687 . . . 4  |-  ( u  =  a  ->  ( E. b  e.  NN0  [. b  /  v ]. ph  <->  E. b  e.  NN0  [. a  /  u ]. [. b  /  v ]. ph )
)
1310, 12syl5bb 249 . . 3  |-  ( u  =  a  ->  ( E. v  e.  NN0  ph  <->  E. b  e.  NN0  [. a  /  u ]. [. b  /  v ]. ph )
)
141, 2, 3, 6, 13cbvrab 2914 . 2  |-  { u  e.  ( NN0  ^m  (
1 ... N ) )  |  E. v  e. 
NN0  ph }  =  {
a  e.  ( NN0 
^m  ( 1 ... N ) )  |  E. b  e.  NN0  [. a  /  u ]. [. b  /  v ]. ph }
15 rexfrabdioph.1 . . 3  |-  M  =  ( N  +  1 )
16 dfsbcq 3123 . . . 4  |-  ( b  =  ( t `  M )  ->  ( [. b  /  v ]. ph  <->  [. ( t `  M )  /  v ]. ph ) )
1716sbcbidv 3175 . . 3  |-  ( b  =  ( t `  M )  ->  ( [. a  /  u ]. [. b  /  v ]. ph  <->  [. a  /  u ]. [. ( t `  M )  /  v ]. ph ) )
18 dfsbcq 3123 . . 3  |-  ( a  =  ( t  |`  ( 1 ... N
) )  ->  ( [. a  /  u ]. [. ( t `  M )  /  v ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. ph ) )
1915, 17, 18rexrabdioph 26744 . 2  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... M ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. ph }  e.  (Dioph `  M ) )  ->  { a  e.  ( NN0  ^m  ( 1 ... N ) )  |  E. b  e. 
NN0  [. a  /  u ]. [. b  /  v ]. ph }  e.  (Dioph `  N ) )
2014, 19syl5eqel 2488 1  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... M ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. ph }  e.  (Dioph `  M ) )  ->  { u  e.  ( NN0  ^m  ( 1 ... N ) )  |  E. v  e.  NN0  ph }  e.  (Dioph `  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667   {crab 2670   [.wsbc 3121    |` cres 4839   ` cfv 5413  (class class class)co 6040    ^m cmap 6977   1c1 8947    + caddc 8949   NN0cn0 10177   ...cfz 10999  Diophcdioph 26703
This theorem is referenced by:  2rexfrabdioph  26746  3rexfrabdioph  26747  7rexfrabdioph  26750  rmxdioph  26977  expdiophlem2  26983
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-hash 11574  df-mzpcl 26670  df-mzp 26671  df-dioph 26704
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