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Theorem mppsval 30282
 Description: Definition of a provable pre-statement, essentially just a reorganization of the arguments of df-mcls . (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mppsval.p mPreSt
mppsval.j mPPSt
mppsval.c mCls
Assertion
Ref Expression
mppsval
Distinct variable groups:   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem mppsval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mppsval.j . 2 mPPSt
2 fveq2 5879 . . . . . . . 8 mPreSt mPreSt
3 mppsval.p . . . . . . . 8 mPreSt
42, 3syl6eqr 2523 . . . . . . 7 mPreSt
54eleq2d 2534 . . . . . 6 mPreSt
6 fveq2 5879 . . . . . . . . 9 mCls mCls
7 mppsval.c . . . . . . . . 9 mCls
86, 7syl6eqr 2523 . . . . . . . 8 mCls
98oveqd 6325 . . . . . . 7 mCls
109eleq2d 2534 . . . . . 6 mCls
115, 10anbi12d 725 . . . . 5 mPreSt mCls
1211oprabbidv 6364 . . . 4 mPreSt mCls
13 df-mpps 30208 . . . 4 mPPSt mPreSt mCls
14 fvex 5889 . . . . . 6 mPreSt
153, 14eqeltri 2545 . . . . 5
163, 1, 7mppspstlem 30281 . . . . 5
1715, 16ssexi 4541 . . . 4
1812, 13, 17fvmpt 5963 . . 3 mPPSt
19 fvprc 5873 . . . 4 mPPSt
20 df-oprab 6312 . . . . 5
21 abn0 3754 . . . . . . 7
22 elfvex 5906 . . . . . . . . . . 11 mPreSt
2322, 3eleq2s 2567 . . . . . . . . . 10
2423ad2antrl 742 . . . . . . . . 9
2524exlimivv 1786 . . . . . . . 8
2625exlimivv 1786 . . . . . . 7
2721, 26sylbi 200 . . . . . 6
2827necon1bi 2671 . . . . 5
2920, 28syl5eq 2517 . . . 4
3019, 29eqtr4d 2508 . . 3 mPPSt
3118, 30pm2.61i 169 . 2 mPPSt
321, 31eqtri 2493 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wa 376   wceq 1452  wex 1671   wcel 1904  cab 2457   wne 2641  cvv 3031  c0 3722  cop 3965  cotp 3967  cfv 5589  (class class class)co 6308  coprab 6309  mPreStcmpst 30183  mClscmcls 30187  mPPStcmpps 30188 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-ot 3968  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpps 30208 This theorem is referenced by:  elmpps  30283  mppspst  30284
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