Theorem vtocl4ga 3570

Description: Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.) (Proof shortened by Wolf Lammen, 31-May-2025.)

Hypotheses

Ref	Expression
vtocl4ga.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
vtocl4ga.2	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
vtocl4ga.3	⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜌))
vtocl4ga.4	⊢ (𝑤 = 𝐷 → (𝜌 ↔ 𝜃))
vtocl4ga.5	⊢ (((𝑥 ∈ 𝑄 ∧ 𝑦 ∈ 𝑅) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇)) → 𝜑)

Assertion

Ref	Expression
vtocl4ga	⊢ (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇)) → 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

Step	Hyp	Ref	Expression
1		vtocl4ga.3	. . . 4 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜌))
2	1	imbi2d 340	. . 3 ⊢ (𝑧 = 𝐶 → (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → 𝜒) ↔ ((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → 𝜌)))
3		vtocl4ga.4	. . . 4 ⊢ (𝑤 = 𝐷 → (𝜌 ↔ 𝜃))
4	3	imbi2d 340	. . 3 ⊢ (𝑤 = 𝐷 → (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → 𝜌) ↔ ((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → 𝜃)))
5		vtocl4ga.1	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
6	5	imbi2d 340	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇) → 𝜑) ↔ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇) → 𝜓)))
7		vtocl4ga.2	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
8	7	imbi2d 340	. . . . 5 ⊢ (𝑦 = 𝐵 → (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇) → 𝜓) ↔ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇) → 𝜒)))
9		vtocl4ga.5	. . . . . 6 ⊢ (((𝑥 ∈ 𝑄 ∧ 𝑦 ∈ 𝑅) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇)) → 𝜑)
10	9	ex 412	. . . . 5 ⊢ ((𝑥 ∈ 𝑄 ∧ 𝑦 ∈ 𝑅) → ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇) → 𝜑))
11	6, 8, 10	vtocl2ga 3562	. . . 4 ⊢ ((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇) → 𝜒))
12	11	com12 32	. . 3 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇) → ((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → 𝜒))
13	2, 4, 12	vtocl2ga 3562	. 2 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇) → ((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → 𝜃))
14	13	impcom 407	1 ⊢ (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇)) → 𝜃)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107

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-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808

This theorem is referenced by:  wrd2ind  14744