Theorem vtocl4ga 3520

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

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		eleq1 2826	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑄 ↔ 𝐴 ∈ 𝑄))
2	1	anbi1d 630	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝑄 ∧ 𝑦 ∈ 𝑅) ↔ (𝐴 ∈ 𝑄 ∧ 𝑦 ∈ 𝑅)))
3	2	anbi1d 630	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑥 ∈ 𝑄 ∧ 𝑦 ∈ 𝑅) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇)) ↔ ((𝐴 ∈ 𝑄 ∧ 𝑦 ∈ 𝑅) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇))))
4		vtocl4ga.1	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	3, 4	imbi12d 345	. . 3 ⊢ (𝑥 = 𝐴 → ((((𝑥 ∈ 𝑄 ∧ 𝑦 ∈ 𝑅) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇)) → 𝜑) ↔ (((𝐴 ∈ 𝑄 ∧ 𝑦 ∈ 𝑅) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇)) → 𝜓)))
6		eleq1 2826	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑅 ↔ 𝐵 ∈ 𝑅))
7	6	anbi2d 629	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝑄 ∧ 𝑦 ∈ 𝑅) ↔ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅)))
8	7	anbi1d 630	. . . 4 ⊢ (𝑦 = 𝐵 → (((𝐴 ∈ 𝑄 ∧ 𝑦 ∈ 𝑅) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇)) ↔ ((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇))))
9		vtocl4ga.2	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
10	8, 9	imbi12d 345	. . 3 ⊢ (𝑦 = 𝐵 → ((((𝐴 ∈ 𝑄 ∧ 𝑦 ∈ 𝑅) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇)) → 𝜓) ↔ (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇)) → 𝜒)))
11		eleq1 2826	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ 𝑆 ↔ 𝐶 ∈ 𝑆))
12	11	anbi1d 630	. . . . 5 ⊢ (𝑧 = 𝐶 → ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇) ↔ (𝐶 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇)))
13	12	anbi2d 629	. . . 4 ⊢ (𝑧 = 𝐶 → (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇)) ↔ ((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝐶 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇))))
14		vtocl4ga.3	. . . 4 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜌))
15	13, 14	imbi12d 345	. . 3 ⊢ (𝑧 = 𝐶 → ((((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇)) → 𝜒) ↔ (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝐶 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇)) → 𝜌)))
16		eleq1 2826	. . . . . 6 ⊢ (𝑤 = 𝐷 → (𝑤 ∈ 𝑇 ↔ 𝐷 ∈ 𝑇))
17	16	anbi2d 629	. . . . 5 ⊢ (𝑤 = 𝐷 → ((𝐶 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇) ↔ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇)))
18	17	anbi2d 629	. . . 4 ⊢ (𝑤 = 𝐷 → (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝐶 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇)) ↔ ((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇))))
19		vtocl4ga.4	. . . 4 ⊢ (𝑤 = 𝐷 → (𝜌 ↔ 𝜃))
20	18, 19	imbi12d 345	. . 3 ⊢ (𝑤 = 𝐷 → ((((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝐶 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇)) → 𝜌) ↔ (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇)) → 𝜃)))
21		vtocl4ga.5	. . 3 ⊢ (((𝑥 ∈ 𝑄 ∧ 𝑦 ∈ 𝑅) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇)) → 𝜑)
22	5, 10, 15, 20, 21	vtocl4g 3519	. 2 ⊢ (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇)) → (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇)) → 𝜃))
23	22	pm2.43i 52	1 ⊢ (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇)) → 𝜃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434

This theorem is referenced by:  wrd2ind  14436