Theorem vtocl4g 3548

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 343	. . 3 ⊢ (𝑧 = 𝐶 → (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → 𝜒) ↔ ((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → 𝜌)))
3		vtocl4g.4	. . . 4 ⊢ (𝑤 = 𝐷 → (𝜌 ↔ 𝜃))
4	3	imbi2d 343	. . 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 3540	. . 3 ⊢ ((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → 𝜒)
9	2, 4, 8	vtocl2g 3540	. 2 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇) → ((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) → 𝜃))
10	9	impcom 413	1 ⊢ (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇)) → 𝜃)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459

This theorem is used by: (None)