Theorem vtocl4g 3569

Description: Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.)

Hypotheses

Ref	Expression
vtocl4g.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
vtocl4g.2	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
vtocl4g.3	⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜌))
vtocl4g.4	⊢ (𝑤 = 𝐷 → (𝜌 ↔ 𝜃))
vtocl4g.5	⊢ 𝜑

Assertion

Ref	Expression
vtocl4g	⊢ (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇)) → 𝜃)

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝑦,𝐵   𝜓,𝑥   𝜒,𝑦   𝑧,𝐶   𝑤,𝐶   𝑤,𝐷   𝑧,𝐴   𝑧,𝑄   𝑧,𝐵   𝑧,𝑅   𝜌,𝑧   𝑤,𝐴   𝑤,𝑄   𝑤,𝐵   𝑤,𝑅   𝜃,𝑤

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)   𝑄(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦,𝑧,𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem vtocl4g

Step	Hyp	Ref	Expression
1		vtocl4g.3	. . . 4 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜌))
2	1	imbi2d 340	. . 3 ⊢ (𝑧 = 𝐶 → (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → 𝜒) ↔ ((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → 𝜌)))
3		vtocl4g.4	. . . 4 ⊢ (𝑤 = 𝐷 → (𝜌 ↔ 𝜃))
4	3	imbi2d 340	. . 3 ⊢ (𝑤 = 𝐷 → (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → 𝜌) ↔ ((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → 𝜃)))
5		vtocl4g.1	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
6		vtocl4g.2	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
7		vtocl4g.5	. . . 4 ⊢ 𝜑
8	5, 6, 7	vtocl2g 3558	. . 3 ⊢ ((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → 𝜒)
9	2, 4, 8	vtocl2g 3558	. 2 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇) → ((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → 𝜃))
10	9	impcom 407	1 ⊢ (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇)) → 𝜃)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466

This theorem is referenced by:  vtocl4gaOLD  3571