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Theorem vtocl3ga 3583
Description: Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.) Reduce axiom usage. (Revised by GG, 3-Oct-2024.) (Proof shortened by Wolf Lammen, 31-May-2025.)
Hypotheses
Ref Expression
vtocl3ga.1 (𝑥 = 𝐴 → (𝜑𝜓))
vtocl3ga.2 (𝑦 = 𝐵 → (𝜓𝜒))
vtocl3ga.3 (𝑧 = 𝐶 → (𝜒𝜃))
vtocl3ga.4 ((𝑥𝐷𝑦𝑅𝑧𝑆) → 𝜑)
Assertion
Ref Expression
vtocl3ga ((𝐴𝐷𝐵𝑅𝐶𝑆) → 𝜃)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑦,𝐵,𝑧   𝑧,𝐶   𝑥,𝐷,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝜓,𝑥   𝜒,𝑦   𝜃,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑦,𝑧)   𝜒(𝑥,𝑧)   𝜃(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem vtocl3ga
StepHypRef Expression
1 vtocl3ga.3 . . . . 5 (𝑧 = 𝐶 → (𝜒𝜃))
21imbi2d 340 . . . 4 (𝑧 = 𝐶 → (((𝐴𝐷𝐵𝑅) → 𝜒) ↔ ((𝐴𝐷𝐵𝑅) → 𝜃)))
3 vtocl3ga.1 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜓))
43imbi2d 340 . . . . . 6 (𝑥 = 𝐴 → ((𝑧𝑆𝜑) ↔ (𝑧𝑆𝜓)))
5 vtocl3ga.2 . . . . . . 7 (𝑦 = 𝐵 → (𝜓𝜒))
65imbi2d 340 . . . . . 6 (𝑦 = 𝐵 → ((𝑧𝑆𝜓) ↔ (𝑧𝑆𝜒)))
7 vtocl3ga.4 . . . . . . 7 ((𝑥𝐷𝑦𝑅𝑧𝑆) → 𝜑)
873expia 1120 . . . . . 6 ((𝑥𝐷𝑦𝑅) → (𝑧𝑆𝜑))
94, 6, 8vtocl2ga 3578 . . . . 5 ((𝐴𝐷𝐵𝑅) → (𝑧𝑆𝜒))
109com12 32 . . . 4 (𝑧𝑆 → ((𝐴𝐷𝐵𝑅) → 𝜒))
112, 10vtoclga 3577 . . 3 (𝐶𝑆 → ((𝐴𝐷𝐵𝑅) → 𝜃))
1211impcom 407 . 2 (((𝐴𝐷𝐵𝑅) ∧ 𝐶𝑆) → 𝜃)
13123impa 1109 1 ((𝐴𝐷𝐵𝑅𝐶𝑆) → 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106
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-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814
This theorem is referenced by:  preq12bg  4858  xpord3pred  8176  jensenlem2  27046  wrdt2ind  32923
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