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Theorem vtocl3gafOLD 3582
Description: Obsolete version of vtocl3gaf 3581 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 2920 . . . . 5 𝑥 𝐴𝑅
8 nfv 1912 . . . . 5 𝑥 𝑦𝑆
9 nfv 1912 . . . . 5 𝑥 𝑧𝑇
107, 8, 9nf3an 1899 . . . 4 𝑥(𝐴𝑅𝑦𝑆𝑧𝑇)
11 vtocl3gaf.1 . . . 4 𝑥𝜓
1210, 11nfim 1894 . . 3 𝑥((𝐴𝑅𝑦𝑆𝑧𝑇) → 𝜓)
132nfel1 2920 . . . . 5 𝑦 𝐴𝑅
144nfel1 2920 . . . . 5 𝑦 𝐵𝑆
15 nfv 1912 . . . . 5 𝑦 𝑧𝑇
1613, 14, 15nf3an 1899 . . . 4 𝑦(𝐴𝑅𝐵𝑆𝑧𝑇)
17 vtocl3gaf.2 . . . 4 𝑦𝜒
1816, 17nfim 1894 . . 3 𝑦((𝐴𝑅𝐵𝑆𝑧𝑇) → 𝜒)
193nfel1 2920 . . . . 5 𝑧 𝐴𝑅
205nfel1 2920 . . . . 5 𝑧 𝐵𝑆
216nfel1 2920 . . . . 5 𝑧 𝐶𝑇
2219, 20, 21nf3an 1899 . . . 4 𝑧(𝐴𝑅𝐵𝑆𝐶𝑇)
23 vtocl3gaf.3 . . . 4 𝑧𝜃
2422, 23nfim 1894 . . 3 𝑧((𝐴𝑅𝐵𝑆𝐶𝑇) → 𝜃)
25 eleq1 2827 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑅𝐴𝑅))
26253anbi1d 1439 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑅𝑦𝑆𝑧𝑇) ↔ (𝐴𝑅𝑦𝑆𝑧𝑇)))
27 vtocl3gaf.4 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
2826, 27imbi12d 344 . . 3 (𝑥 = 𝐴 → (((𝑥𝑅𝑦𝑆𝑧𝑇) → 𝜑) ↔ ((𝐴𝑅𝑦𝑆𝑧𝑇) → 𝜓)))
29 eleq1 2827 . . . . 5 (𝑦 = 𝐵 → (𝑦𝑆𝐵𝑆))
30293anbi2d 1440 . . . 4 (𝑦 = 𝐵 → ((𝐴𝑅𝑦𝑆𝑧𝑇) ↔ (𝐴𝑅𝐵𝑆𝑧𝑇)))
31 vtocl3gaf.5 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
3230, 31imbi12d 344 . . 3 (𝑦 = 𝐵 → (((𝐴𝑅𝑦𝑆𝑧𝑇) → 𝜓) ↔ ((𝐴𝑅𝐵𝑆𝑧𝑇) → 𝜒)))
33 eleq1 2827 . . . . 5 (𝑧 = 𝐶 → (𝑧𝑇𝐶𝑇))
34333anbi3d 1441 . . . 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 3573 . 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 1537  wnf 1780  wcel 2106  wnfc 2888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-v 3480
This theorem is referenced by: (None)
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