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Theorem vtocl4ga 3418
Description: Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.)
Hypotheses
Ref Expression
vtocl4ga.1 (𝑥 = 𝐴 → (𝜑𝜓))
vtocl4ga.2 (𝑦 = 𝐵 → (𝜓𝜒))
vtocl4ga.3 (𝑧 = 𝐶 → (𝜒𝜌))
vtocl4ga.4 (𝑤 = 𝐷 → (𝜌𝜃))
vtocl4ga.5 (((𝑥𝑄𝑦𝑅) ∧ (𝑧𝑆𝑤𝑇)) → 𝜑)
Assertion
Ref Expression
vtocl4ga (((𝐴𝑄𝐵𝑅) ∧ (𝐶𝑆𝐷𝑇)) → 𝜃)
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝐵,𝑦,𝑧   𝑤,𝐶,𝑧   𝑤,𝐷   𝑤,𝑅,𝑥,𝑦,𝑧   𝑤,𝑆,𝑥,𝑦,𝑧   𝑤,𝑇,𝑥,𝑦,𝑧   𝑤,𝑄,𝑥,𝑦,𝑧   𝜓,𝑥   𝜌,𝑧   𝜃,𝑤   𝜒,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑦,𝑧,𝑤)   𝜒(𝑥,𝑧,𝑤)   𝜃(𝑥,𝑦,𝑧)   𝜌(𝑥,𝑦,𝑤)   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦,𝑧)

Proof of Theorem vtocl4ga
StepHypRef Expression
1 eleq1 2827 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑄𝐴𝑄))
21anbi1d 743 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝑄𝑦𝑅) ↔ (𝐴𝑄𝑦𝑅)))
32anbi1d 743 . . . 4 (𝑥 = 𝐴 → (((𝑥𝑄𝑦𝑅) ∧ (𝑧𝑆𝑤𝑇)) ↔ ((𝐴𝑄𝑦𝑅) ∧ (𝑧𝑆𝑤𝑇))))
4 vtocl4ga.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
53, 4imbi12d 333 . . 3 (𝑥 = 𝐴 → ((((𝑥𝑄𝑦𝑅) ∧ (𝑧𝑆𝑤𝑇)) → 𝜑) ↔ (((𝐴𝑄𝑦𝑅) ∧ (𝑧𝑆𝑤𝑇)) → 𝜓)))
6 eleq1 2827 . . . . . 6 (𝑦 = 𝐵 → (𝑦𝑅𝐵𝑅))
76anbi2d 742 . . . . 5 (𝑦 = 𝐵 → ((𝐴𝑄𝑦𝑅) ↔ (𝐴𝑄𝐵𝑅)))
87anbi1d 743 . . . 4 (𝑦 = 𝐵 → (((𝐴𝑄𝑦𝑅) ∧ (𝑧𝑆𝑤𝑇)) ↔ ((𝐴𝑄𝐵𝑅) ∧ (𝑧𝑆𝑤𝑇))))
9 vtocl4ga.2 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
108, 9imbi12d 333 . . 3 (𝑦 = 𝐵 → ((((𝐴𝑄𝑦𝑅) ∧ (𝑧𝑆𝑤𝑇)) → 𝜓) ↔ (((𝐴𝑄𝐵𝑅) ∧ (𝑧𝑆𝑤𝑇)) → 𝜒)))
11 eleq1 2827 . . . . . 6 (𝑧 = 𝐶 → (𝑧𝑆𝐶𝑆))
1211anbi1d 743 . . . . 5 (𝑧 = 𝐶 → ((𝑧𝑆𝑤𝑇) ↔ (𝐶𝑆𝑤𝑇)))
1312anbi2d 742 . . . 4 (𝑧 = 𝐶 → (((𝐴𝑄𝐵𝑅) ∧ (𝑧𝑆𝑤𝑇)) ↔ ((𝐴𝑄𝐵𝑅) ∧ (𝐶𝑆𝑤𝑇))))
14 vtocl4ga.3 . . . 4 (𝑧 = 𝐶 → (𝜒𝜌))
1513, 14imbi12d 333 . . 3 (𝑧 = 𝐶 → ((((𝐴𝑄𝐵𝑅) ∧ (𝑧𝑆𝑤𝑇)) → 𝜒) ↔ (((𝐴𝑄𝐵𝑅) ∧ (𝐶𝑆𝑤𝑇)) → 𝜌)))
16 eleq1 2827 . . . . . 6 (𝑤 = 𝐷 → (𝑤𝑇𝐷𝑇))
1716anbi2d 742 . . . . 5 (𝑤 = 𝐷 → ((𝐶𝑆𝑤𝑇) ↔ (𝐶𝑆𝐷𝑇)))
1817anbi2d 742 . . . 4 (𝑤 = 𝐷 → (((𝐴𝑄𝐵𝑅) ∧ (𝐶𝑆𝑤𝑇)) ↔ ((𝐴𝑄𝐵𝑅) ∧ (𝐶𝑆𝐷𝑇))))
19 vtocl4ga.4 . . . 4 (𝑤 = 𝐷 → (𝜌𝜃))
2018, 19imbi12d 333 . . 3 (𝑤 = 𝐷 → ((((𝐴𝑄𝐵𝑅) ∧ (𝐶𝑆𝑤𝑇)) → 𝜌) ↔ (((𝐴𝑄𝐵𝑅) ∧ (𝐶𝑆𝐷𝑇)) → 𝜃)))
21 vtocl4ga.5 . . 3 (((𝑥𝑄𝑦𝑅) ∧ (𝑧𝑆𝑤𝑇)) → 𝜑)
225, 10, 15, 20, 21vtocl4g 3417 . 2 (((𝐴𝑄𝐵𝑅) ∧ (𝐶𝑆𝐷𝑇)) → (((𝐴𝑄𝐵𝑅) ∧ (𝐶𝑆𝐷𝑇)) → 𝜃))
2322pm2.43i 52 1 (((𝐴𝑄𝐵𝑅) ∧ (𝐶𝑆𝐷𝑇)) → 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342
This theorem is referenced by:  wrd2ind  13697
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