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Theorem vtocl3ga 3540
Description: Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.) Reduce axiom usage. (Revised by Gino Giotto, 3-Oct-2024.)
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 eleq1 2822 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐷𝐴𝐷))
213anbi1d 1441 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐷𝑦𝑅𝑧𝑆) ↔ (𝐴𝐷𝑦𝑅𝑧𝑆)))
3 vtocl3ga.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
42, 3imbi12d 345 . . 3 (𝑥 = 𝐴 → (((𝑥𝐷𝑦𝑅𝑧𝑆) → 𝜑) ↔ ((𝐴𝐷𝑦𝑅𝑧𝑆) → 𝜓)))
5 eleq1 2822 . . . . 5 (𝑦 = 𝐵 → (𝑦𝑅𝐵𝑅))
653anbi2d 1442 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐷𝑦𝑅𝑧𝑆) ↔ (𝐴𝐷𝐵𝑅𝑧𝑆)))
7 vtocl3ga.2 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
86, 7imbi12d 345 . . 3 (𝑦 = 𝐵 → (((𝐴𝐷𝑦𝑅𝑧𝑆) → 𝜓) ↔ ((𝐴𝐷𝐵𝑅𝑧𝑆) → 𝜒)))
9 eleq1 2822 . . . . 5 (𝑧 = 𝐶 → (𝑧𝑆𝐶𝑆))
1093anbi3d 1443 . . . 4 (𝑧 = 𝐶 → ((𝐴𝐷𝐵𝑅𝑧𝑆) ↔ (𝐴𝐷𝐵𝑅𝐶𝑆)))
11 vtocl3ga.3 . . . 4 (𝑧 = 𝐶 → (𝜒𝜃))
1210, 11imbi12d 345 . . 3 (𝑧 = 𝐶 → (((𝐴𝐷𝐵𝑅𝑧𝑆) → 𝜒) ↔ ((𝐴𝐷𝐵𝑅𝐶𝑆) → 𝜃)))
13 vtocl3ga.4 . . 3 ((𝑥𝐷𝑦𝑅𝑧𝑆) → 𝜑)
144, 8, 12, 13vtocl3g 3534 . 2 ((𝐴𝐷𝐵𝑅𝐶𝑆) → ((𝐴𝐷𝐵𝑅𝐶𝑆) → 𝜃))
1514pm2.43i 52 1 ((𝐴𝐷𝐵𝑅𝐶𝑆) → 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1088   = wceq 1542  wcel 2107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3449
This theorem is referenced by:  preq12bg  4815  poclOLD  5557  xpord3pred  8088  jensenlem2  26360  wrdt2ind  31863
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