Theorem rspc4v 3630

Description: 4-variable restricted specialization, using implicit substitution. (Contributed by Scott Fenton, 7-Feb-2025.)

Hypotheses

Ref	Expression
rspc4v.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))
rspc4v.2	⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))
rspc4v.3	⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜏))
rspc4v.4	⊢ (𝑤 = 𝐷 → (𝜏 ↔ 𝜓))

Assertion

Ref	Expression
rspc4v	⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈)) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 ∀𝑤 ∈ 𝑈 𝜑 → 𝜓))

Distinct variable groups:   𝑤,𝐴,𝑥,𝑦,𝑧   𝑤,𝐵,𝑦,𝑧   𝑤,𝐶,𝑧   𝑤,𝐷   𝑥,𝑅   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦,𝑧   𝑤,𝑈,𝑥,𝑦,𝑧   𝜒,𝑥   𝜓,𝑤   𝜏,𝑧   𝜃,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑦,𝑧,𝑤)   𝜃(𝑥,𝑧,𝑤)   𝜏(𝑥,𝑦,𝑤)   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦,𝑧)   𝑅(𝑦,𝑧,𝑤)   𝑆(𝑧,𝑤)   𝑇(𝑤)

Proof of Theorem rspc4v

Step	Hyp	Ref	Expression
1		df-3an 1088	. . 3 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑇))
2		rspc4v.1	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))
3	2	ralbidv 3176	. . . . 5 ⊢ (𝑥 = 𝐴 → (∀𝑤 ∈ 𝑈 𝜑 ↔ ∀𝑤 ∈ 𝑈 𝜒))
4		rspc4v.2	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))
5	4	ralbidv 3176	. . . . 5 ⊢ (𝑦 = 𝐵 → (∀𝑤 ∈ 𝑈 𝜒 ↔ ∀𝑤 ∈ 𝑈 𝜃))
6		rspc4v.3	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜏))
7	6	ralbidv 3176	. . . . 5 ⊢ (𝑧 = 𝐶 → (∀𝑤 ∈ 𝑈 𝜃 ↔ ∀𝑤 ∈ 𝑈 𝜏))
8	3, 5, 7	rspc3v 3627	. . . 4 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 ∀𝑤 ∈ 𝑈 𝜑 → ∀𝑤 ∈ 𝑈 𝜏))
9		rspc4v.4	. . . . 5 ⊢ (𝑤 = 𝐷 → (𝜏 ↔ 𝜓))
10	9	rspcv 3608	. . . 4 ⊢ (𝐷 ∈ 𝑈 → (∀𝑤 ∈ 𝑈 𝜏 → 𝜓))
11	8, 10	sylan9 507	. . 3 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝐷 ∈ 𝑈) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 ∀𝑤 ∈ 𝑈 𝜑 → 𝜓))
12	1, 11	sylanbr 581	. 2 ⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑇) ∧ 𝐷 ∈ 𝑈) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 ∀𝑤 ∈ 𝑈 𝜑 → 𝜓))
13	12	anasss 466	1 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈)) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 ∀𝑤 ∈ 𝑈 𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1088  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061

This theorem is referenced by:  rspc6v  3631  rspc8v  3632