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Theorem vtocl3gaf 2616
Description: Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.)
Hypotheses
Ref Expression
vtocl3gaf.a  F/_
vtocl3gaf.b  F/_
vtocl3gaf.c  F/_
vtocl3gaf.d  F/_
vtocl3gaf.e  F/_
vtocl3gaf.f  F/_ C
vtocl3gaf.1  F/
vtocl3gaf.2  F/
vtocl3gaf.3  F/
vtocl3gaf.4
vtocl3gaf.5
vtocl3gaf.6  C
vtocl3gaf.7  R  S  T
Assertion
Ref Expression
vtocl3gaf  R  S  C  T
Distinct variable groups:   ,,, R   , S,,   , T,,
Allowed substitution hints:   (,,)   (,,)   (,,)   (,,)   (,,)   (,,)    C(,,)

Proof of Theorem vtocl3gaf
StepHypRef Expression
1 vtocl3gaf.a . . 3  F/_
2 vtocl3gaf.b . . 3  F/_
3 vtocl3gaf.c . . 3  F/_
4 vtocl3gaf.d . . 3  F/_
5 vtocl3gaf.e . . 3  F/_
6 vtocl3gaf.f . . 3  F/_ C
71nfel1 2185 . . . . 5  F/  R
8 nfv 1418 . . . . 5  F/  S
9 nfv 1418 . . . . 5  F/  T
107, 8, 9nf3an 1455 . . . 4  F/  R  S  T
11 vtocl3gaf.1 . . . 4  F/
1210, 11nfim 1461 . . 3  F/  R  S  T
132nfel1 2185 . . . . 5  F/  R
144nfel1 2185 . . . . 5  F/  S
15 nfv 1418 . . . . 5  F/  T
1613, 14, 15nf3an 1455 . . . 4  F/  R  S  T
17 vtocl3gaf.2 . . . 4  F/
1816, 17nfim 1461 . . 3  F/  R  S  T
193nfel1 2185 . . . . 5  F/  R
205nfel1 2185 . . . . 5  F/  S
216nfel1 2185 . . . . 5  F/  C  T
2219, 20, 21nf3an 1455 . . . 4  F/  R  S  C  T
23 vtocl3gaf.3 . . . 4  F/
2422, 23nfim 1461 . . 3  F/  R  S  C  T
25 eleq1 2097 . . . . 5  R  R
26253anbi1d 1210 . . . 4  R  S  T  R  S  T
27 vtocl3gaf.4 . . . 4
2826, 27imbi12d 223 . . 3  R  S  T  R  S  T
29 eleq1 2097 . . . . 5  S  S
30293anbi2d 1211 . . . 4  R  S  T  R  S  T
31 vtocl3gaf.5 . . . 4
3230, 31imbi12d 223 . . 3  R  S  T  R  S  T
33 eleq1 2097 . . . . 5  C  T  C  T
34333anbi3d 1212 . . . 4  C  R  S  T  R  S  C  T
35 vtocl3gaf.6 . . . 4  C
3634, 35imbi12d 223 . . 3  C  R  S  T  R  S  C  T
37 vtocl3gaf.7 . . 3  R  S  T
381, 2, 3, 4, 5, 6, 12, 18, 24, 28, 32, 36, 37vtocl3gf 2610 . 2  R  S  C  T  R  S  C  T
3938pm2.43i 43 1  R  S  C  T
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98   w3a 884   wceq 1242   F/wnf 1346   wcel 1390   F/_wnfc 2162
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-bndl 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-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553
This theorem is referenced by:  vtocl3ga  2617
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