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Theorem vtocl3gaf 3427
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 2922 . . . . 5 𝑥 𝐴𝑅
8 nfv 2009 . . . . 5 𝑥 𝑦𝑆
9 nfv 2009 . . . . 5 𝑥 𝑧𝑇
107, 8, 9nf3an 2000 . . . 4 𝑥(𝐴𝑅𝑦𝑆𝑧𝑇)
11 vtocl3gaf.1 . . . 4 𝑥𝜓
1210, 11nfim 1995 . . 3 𝑥((𝐴𝑅𝑦𝑆𝑧𝑇) → 𝜓)
132nfel1 2922 . . . . 5 𝑦 𝐴𝑅
144nfel1 2922 . . . . 5 𝑦 𝐵𝑆
15 nfv 2009 . . . . 5 𝑦 𝑧𝑇
1613, 14, 15nf3an 2000 . . . 4 𝑦(𝐴𝑅𝐵𝑆𝑧𝑇)
17 vtocl3gaf.2 . . . 4 𝑦𝜒
1816, 17nfim 1995 . . 3 𝑦((𝐴𝑅𝐵𝑆𝑧𝑇) → 𝜒)
193nfel1 2922 . . . . 5 𝑧 𝐴𝑅
205nfel1 2922 . . . . 5 𝑧 𝐵𝑆
216nfel1 2922 . . . . 5 𝑧 𝐶𝑇
2219, 20, 21nf3an 2000 . . . 4 𝑧(𝐴𝑅𝐵𝑆𝐶𝑇)
23 vtocl3gaf.3 . . . 4 𝑧𝜃
2422, 23nfim 1995 . . 3 𝑧((𝐴𝑅𝐵𝑆𝐶𝑇) → 𝜃)
25 eleq1 2832 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑅𝐴𝑅))
26253anbi1d 1564 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑅𝑦𝑆𝑧𝑇) ↔ (𝐴𝑅𝑦𝑆𝑧𝑇)))
27 vtocl3gaf.4 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
2826, 27imbi12d 335 . . 3 (𝑥 = 𝐴 → (((𝑥𝑅𝑦𝑆𝑧𝑇) → 𝜑) ↔ ((𝐴𝑅𝑦𝑆𝑧𝑇) → 𝜓)))
29 eleq1 2832 . . . . 5 (𝑦 = 𝐵 → (𝑦𝑆𝐵𝑆))
30293anbi2d 1565 . . . 4 (𝑦 = 𝐵 → ((𝐴𝑅𝑦𝑆𝑧𝑇) ↔ (𝐴𝑅𝐵𝑆𝑧𝑇)))
31 vtocl3gaf.5 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
3230, 31imbi12d 335 . . 3 (𝑦 = 𝐵 → (((𝐴𝑅𝑦𝑆𝑧𝑇) → 𝜓) ↔ ((𝐴𝑅𝐵𝑆𝑧𝑇) → 𝜒)))
33 eleq1 2832 . . . . 5 (𝑧 = 𝐶 → (𝑧𝑇𝐶𝑇))
34333anbi3d 1566 . . . 4 (𝑧 = 𝐶 → ((𝐴𝑅𝐵𝑆𝑧𝑇) ↔ (𝐴𝑅𝐵𝑆𝐶𝑇)))
35 vtocl3gaf.6 . . . 4 (𝑧 = 𝐶 → (𝜒𝜃))
3634, 35imbi12d 335 . . 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 3421 . 2 ((𝐴𝑅𝐵𝑆𝐶𝑇) → ((𝐴𝑅𝐵𝑆𝐶𝑇) → 𝜃))
3938pm2.43i 52 1 ((𝐴𝑅𝐵𝑆𝐶𝑇) → 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  w3a 1107   = wceq 1652  wnf 1878  wcel 2155  wnfc 2894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352
This theorem is referenced by:  vtocl3ga  3428
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