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Theorem vtocl3gaf 2639
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 𝑥𝐴
vtocl3gaf.b 𝑦𝐴
vtocl3gaf.c 𝑧𝐴
vtocl3gaf.d 𝑦𝐵
vtocl3gaf.e 𝑧𝐵
vtocl3gaf.f 𝑧𝐶
vtocl3gaf.1 𝑥𝜓
vtocl3gaf.2 𝑦𝜒
vtocl3gaf.3 𝑧𝜃
vtocl3gaf.4 (𝑥 = 𝐴 → (𝜑𝜓))
vtocl3gaf.5 (𝑦 = 𝐵 → (𝜓𝜒))
vtocl3gaf.6 (𝑧 = 𝐶 → (𝜒𝜃))
vtocl3gaf.7 ((𝑥𝑅𝑦𝑆𝑧𝑇) → 𝜑)
Assertion
Ref Expression
vtocl3gaf ((𝐴𝑅𝐵𝑆𝐶𝑇) → 𝜃)
Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝑆,𝑦,𝑧   𝑥,𝑇,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑥,𝑦,𝑧)   𝜃(𝑥,𝑦,𝑧)   𝐴(𝑥,𝑦,𝑧)   𝐵(𝑥,𝑦,𝑧)   𝐶(𝑥,𝑦,𝑧)

Proof of Theorem vtocl3gaf
StepHypRef Expression
1 vtocl3gaf.a . . 3 𝑥𝐴
2 vtocl3gaf.b . . 3 𝑦𝐴
3 vtocl3gaf.c . . 3 𝑧𝐴
4 vtocl3gaf.d . . 3 𝑦𝐵
5 vtocl3gaf.e . . 3 𝑧𝐵
6 vtocl3gaf.f . . 3 𝑧𝐶
71nfel1 2204 . . . . 5 𝑥 𝐴𝑅
8 nfv 1437 . . . . 5 𝑥 𝑦𝑆
9 nfv 1437 . . . . 5 𝑥 𝑧𝑇
107, 8, 9nf3an 1474 . . . 4 𝑥(𝐴𝑅𝑦𝑆𝑧𝑇)
11 vtocl3gaf.1 . . . 4 𝑥𝜓
1210, 11nfim 1480 . . 3 𝑥((𝐴𝑅𝑦𝑆𝑧𝑇) → 𝜓)
132nfel1 2204 . . . . 5 𝑦 𝐴𝑅
144nfel1 2204 . . . . 5 𝑦 𝐵𝑆
15 nfv 1437 . . . . 5 𝑦 𝑧𝑇
1613, 14, 15nf3an 1474 . . . 4 𝑦(𝐴𝑅𝐵𝑆𝑧𝑇)
17 vtocl3gaf.2 . . . 4 𝑦𝜒
1816, 17nfim 1480 . . 3 𝑦((𝐴𝑅𝐵𝑆𝑧𝑇) → 𝜒)
193nfel1 2204 . . . . 5 𝑧 𝐴𝑅
205nfel1 2204 . . . . 5 𝑧 𝐵𝑆
216nfel1 2204 . . . . 5 𝑧 𝐶𝑇
2219, 20, 21nf3an 1474 . . . 4 𝑧(𝐴𝑅𝐵𝑆𝐶𝑇)
23 vtocl3gaf.3 . . . 4 𝑧𝜃
2422, 23nfim 1480 . . 3 𝑧((𝐴𝑅𝐵𝑆𝐶𝑇) → 𝜃)
25 eleq1 2116 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑅𝐴𝑅))
26253anbi1d 1222 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑅𝑦𝑆𝑧𝑇) ↔ (𝐴𝑅𝑦𝑆𝑧𝑇)))
27 vtocl3gaf.4 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
2826, 27imbi12d 227 . . 3 (𝑥 = 𝐴 → (((𝑥𝑅𝑦𝑆𝑧𝑇) → 𝜑) ↔ ((𝐴𝑅𝑦𝑆𝑧𝑇) → 𝜓)))
29 eleq1 2116 . . . . 5 (𝑦 = 𝐵 → (𝑦𝑆𝐵𝑆))
30293anbi2d 1223 . . . 4 (𝑦 = 𝐵 → ((𝐴𝑅𝑦𝑆𝑧𝑇) ↔ (𝐴𝑅𝐵𝑆𝑧𝑇)))
31 vtocl3gaf.5 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
3230, 31imbi12d 227 . . 3 (𝑦 = 𝐵 → (((𝐴𝑅𝑦𝑆𝑧𝑇) → 𝜓) ↔ ((𝐴𝑅𝐵𝑆𝑧𝑇) → 𝜒)))
33 eleq1 2116 . . . . 5 (𝑧 = 𝐶 → (𝑧𝑇𝐶𝑇))
34333anbi3d 1224 . . . 4 (𝑧 = 𝐶 → ((𝐴𝑅𝐵𝑆𝑧𝑇) ↔ (𝐴𝑅𝐵𝑆𝐶𝑇)))
35 vtocl3gaf.6 . . . 4 (𝑧 = 𝐶 → (𝜒𝜃))
3634, 35imbi12d 227 . . 3 (𝑧 = 𝐶 → (((𝐴𝑅𝐵𝑆𝑧𝑇) → 𝜒) ↔ ((𝐴𝑅𝐵𝑆𝐶𝑇) → 𝜃)))
37 vtocl3gaf.7 . . 3 ((𝑥𝑅𝑦𝑆𝑧𝑇) → 𝜑)
381, 2, 3, 4, 5, 6, 12, 18, 24, 28, 32, 36, 37vtocl3gf 2633 . 2 ((𝐴𝑅𝐵𝑆𝐶𝑇) → ((𝐴𝑅𝐵𝑆𝐶𝑇) → 𝜃))
3938pm2.43i 47 1 ((𝐴𝑅𝐵𝑆𝐶𝑇) → 𝜃)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  w3a 896   = wceq 1259  wnf 1365  wcel 1409  wnfc 2181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576
This theorem is referenced by:  vtocl3ga  2640
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