Theorem rspc3dv 3627

Description: 3-variable restricted specialization, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2025.)

Hypotheses

Ref	Expression
rspc3dv.1	⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))
rspc3dv.2	⊢ (𝑦 = 𝐵 → (𝜃 ↔ 𝜏))
rspc3dv.3	⊢ (𝑧 = 𝐶 → (𝜏 ↔ 𝜒))
rspc3dv.4	⊢ (𝜑 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐸 ∀𝑧 ∈ 𝐹 𝜓)
rspc3dv.5	⊢ (𝜑 → 𝐴 ∈ 𝐷)
rspc3dv.6	⊢ (𝜑 → 𝐵 ∈ 𝐸)
rspc3dv.7	⊢ (𝜑 → 𝐶 ∈ 𝐹)

Assertion

Ref	Expression
rspc3dv	⊢ (𝜑 → 𝜒)

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑦,𝐵,𝑧   𝑧,𝐶   𝑥,𝐷   𝑥,𝐸,𝑦   𝑥,𝐹,𝑦,𝑧   𝜒,𝑧   𝜏,𝑦   𝜃,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑥,𝑦)   𝜃(𝑦,𝑧)   𝜏(𝑥,𝑧)   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐷(𝑦,𝑧)   𝐸(𝑧)

Proof of Theorem rspc3dv

Step	Hyp	Ref	Expression
1		rspc3dv.5	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
2		rspc3dv.6	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝐸)
3		rspc3dv.7	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐹)
4	1, 2, 3	3jca 1125	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹))
5		rspc3dv.4	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐸 ∀𝑧 ∈ 𝐹 𝜓)
6		rspc3dv.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))
7		rspc3dv.2	. . 3 ⊢ (𝑦 = 𝐵 → (𝜃 ↔ 𝜏))
8		rspc3dv.3	. . 3 ⊢ (𝑧 = 𝐶 → (𝜏 ↔ 𝜒))
9	6, 7, 8	rspc3v 3625	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹) → (∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐸 ∀𝑧 ∈ 𝐹 𝜓 → 𝜒))
10	4, 5, 9	sylc 65	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3058

This theorem is referenced by:  mulsasslem3  28083  domnlcan  32966