MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vtocl3gafOLD Structured version   Visualization version   GIF version

Theorem vtocl3gafOLD 3581
Description: Obsolete version of vtocl3gaf 3580 as of 31-May-2025. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
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
vtocl3gafOLD ((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 vtocl3gafOLD
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 2921 . . . . 5 𝑥 𝐴𝑅
8 nfv 1913 . . . . 5 𝑥 𝑦𝑆
9 nfv 1913 . . . . 5 𝑥 𝑧𝑇
107, 8, 9nf3an 1900 . . . 4 𝑥(𝐴𝑅𝑦𝑆𝑧𝑇)
11 vtocl3gaf.1 . . . 4 𝑥𝜓
1210, 11nfim 1895 . . 3 𝑥((𝐴𝑅𝑦𝑆𝑧𝑇) → 𝜓)
132nfel1 2921 . . . . 5 𝑦 𝐴𝑅
144nfel1 2921 . . . . 5 𝑦 𝐵𝑆
15 nfv 1913 . . . . 5 𝑦 𝑧𝑇
1613, 14, 15nf3an 1900 . . . 4 𝑦(𝐴𝑅𝐵𝑆𝑧𝑇)
17 vtocl3gaf.2 . . . 4 𝑦𝜒
1816, 17nfim 1895 . . 3 𝑦((𝐴𝑅𝐵𝑆𝑧𝑇) → 𝜒)
193nfel1 2921 . . . . 5 𝑧 𝐴𝑅
205nfel1 2921 . . . . 5 𝑧 𝐵𝑆
216nfel1 2921 . . . . 5 𝑧 𝐶𝑇
2219, 20, 21nf3an 1900 . . . 4 𝑧(𝐴𝑅𝐵𝑆𝐶𝑇)
23 vtocl3gaf.3 . . . 4 𝑧𝜃
2422, 23nfim 1895 . . 3 𝑧((𝐴𝑅𝐵𝑆𝐶𝑇) → 𝜃)
25 eleq1 2828 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑅𝐴𝑅))
26253anbi1d 1441 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑅𝑦𝑆𝑧𝑇) ↔ (𝐴𝑅𝑦𝑆𝑧𝑇)))
27 vtocl3gaf.4 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
2826, 27imbi12d 344 . . 3 (𝑥 = 𝐴 → (((𝑥𝑅𝑦𝑆𝑧𝑇) → 𝜑) ↔ ((𝐴𝑅𝑦𝑆𝑧𝑇) → 𝜓)))
29 eleq1 2828 . . . . 5 (𝑦 = 𝐵 → (𝑦𝑆𝐵𝑆))
30293anbi2d 1442 . . . 4 (𝑦 = 𝐵 → ((𝐴𝑅𝑦𝑆𝑧𝑇) ↔ (𝐴𝑅𝐵𝑆𝑧𝑇)))
31 vtocl3gaf.5 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
3230, 31imbi12d 344 . . 3 (𝑦 = 𝐵 → (((𝐴𝑅𝑦𝑆𝑧𝑇) → 𝜓) ↔ ((𝐴𝑅𝐵𝑆𝑧𝑇) → 𝜒)))
33 eleq1 2828 . . . . 5 (𝑧 = 𝐶 → (𝑧𝑇𝐶𝑇))
34333anbi3d 1443 . . . 4 (𝑧 = 𝐶 → ((𝐴𝑅𝐵𝑆𝑧𝑇) ↔ (𝐴𝑅𝐵𝑆𝐶𝑇)))
35 vtocl3gaf.6 . . . 4 (𝑧 = 𝐶 → (𝜒𝜃))
3634, 35imbi12d 344 . . 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 3572 . 2 ((𝐴𝑅𝐵𝑆𝐶𝑇) → ((𝐴𝑅𝐵𝑆𝐶𝑇) → 𝜃))
3938pm2.43i 52 1 ((𝐴𝑅𝐵𝑆𝐶𝑇) → 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086   = wceq 1539  wnf 1782  wcel 2107  wnfc 2889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-v 3481
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator