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Theorem riota5f 5435
Description: A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
riota5f.1  F/_
riota5f.2
riota5f.3
Assertion
Ref Expression
riota5f  iota_
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem riota5f
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 riota5f.3 . . 3
21ralrimiva 2386 . 2
3 riota5f.2 . . . 4
4 a1tru 1258 . . . . . . 7
5 reu6i 2726 . . . . . . . . 9
65adantl 262 . . . . . . . 8
7 nfv 1418 . . . . . . . . . 10  F/
8 nfv 1418 . . . . . . . . . . 11  F/
9 nfra1 2349 . . . . . . . . . . 11  F/
108, 9nfan 1454 . . . . . . . . . 10  F/
117, 10nfan 1454 . . . . . . . . 9  F/
12 nfcvd 2176 . . . . . . . . 9  F/_
13 nfvd 1419 . . . . . . . . 9  F/
14 simprl 483 . . . . . . . . 9
15 simpr 103 . . . . . . . . . . 11
16 simplrr 488 . . . . . . . . . . . 12
17 simplrl 487 . . . . . . . . . . . . 13
1815, 17eqeltrd 2111 . . . . . . . . . . . 12
19 rsp 2363 . . . . . . . . . . . 12
2016, 18, 19sylc 56 . . . . . . . . . . 11
2115, 20mpbird 156 . . . . . . . . . 10
22 a1tru 1258 . . . . . . . . . 10
2321, 222thd 164 . . . . . . . . 9
2411, 12, 13, 14, 23riota2df 5431 . . . . . . . 8  iota_
256, 24mpdan 398 . . . . . . 7  iota_
264, 25mpbid 135 . . . . . 6  iota_
2726expr 357 . . . . 5  iota_
2827ralrimiva 2386 . . . 4  iota_
29 rspsbc 2834 . . . 4  iota_  [.  ].  iota_
303, 28, 29sylc 56 . . 3  [.  ].  iota_
31 nfcvd 2176 . . . . . . . 8  F/_
32 riota5f.1 . . . . . . . 8  F/_
3331, 32nfeqd 2189 . . . . . . 7  F/
347, 33nfan1 1453 . . . . . 6  F/
35 simpr 103 . . . . . . . 8
3635eqeq2d 2048 . . . . . . 7
3736bibi2d 221 . . . . . 6
3834, 37ralbid 2318 . . . . 5
3935eqeq2d 2048 . . . . 5  iota_  iota_
4038, 39imbi12d 223 . . . 4  iota_  iota_
413, 40sbcied 2793 . . 3  [.  ].  iota_  iota_
4230, 41mpbid 135 . 2  iota_
432, 42mpd 13 1  iota_
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wtru 1243   wcel 1390   F/_wnfc 2162  wral 2300  wreu 2302   [.wsbc 2758   iota_crio 5410
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-v 2553  df-sbc 2759  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810  df-riota 5411
This theorem is referenced by:  riota5  5436
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