Theorem rspc4v 3595

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 1105	. . 3 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑇))
2		rspc4v.1	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))
3	2	ralbidv 3185	. . . . 5 ⊢ (𝑥 = 𝐴 → (∀𝑤 ∈ 𝑈 𝜑 ↔ ∀𝑤 ∈ 𝑈 𝜒))
4		rspc4v.2	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))
5	4	ralbidv 3185	. . . . 5 ⊢ (𝑦 = 𝐵 → (∀𝑤 ∈ 𝑈 𝜒 ↔ ∀𝑤 ∈ 𝑈 𝜃))
6		rspc4v.3	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜏))
7	6	ralbidv 3185	. . . . 5 ⊢ (𝑧 = 𝐶 → (∀𝑤 ∈ 𝑈 𝜃 ↔ ∀𝑤 ∈ 𝑈 𝜏))
8	3, 5, 7	rspc3v 3591	. . . 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 3572	. . . 4 ⊢ (𝐷 ∈ 𝑈 → (∀𝑤 ∈ 𝑈 𝜏 → 𝜓))
11	8, 10	sylan9 517	. . 3 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝐷 ∈ 𝑈) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 ∀𝑤 ∈ 𝑈 𝜑 → 𝜓))
12	1, 11	sylanbr 594	. 2 ⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑇) ∧ 𝐷 ∈ 𝑈) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 ∀𝑤 ∈ 𝑈 𝜑 → 𝜓))
13	12	anasss 472	1 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈)) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 ∀𝑤 ∈ 𝑈 𝜑 → 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077

This theorem is used by:  rspc6v  3596  rspc8v  3597