Theorem 3rspcedvd 40110

Description: Triple application of rspcedvd 3555. (Contributed by Steven Nguyen, 27-Feb-2023.)

Hypotheses

Ref	Expression
3rspcedvd.a	⊢ (𝜑 → 𝐴 ∈ 𝐷)
3rspcedvd.b	⊢ (𝜑 → 𝐵 ∈ 𝐷)
3rspcedvd.c	⊢ (𝜑 → 𝐶 ∈ 𝐷)
3rspcedvd.1	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
3rspcedvd.2	⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (𝜒 ↔ 𝜃))
3rspcedvd.3	⊢ ((𝜑 ∧ 𝑧 = 𝐶) → (𝜃 ↔ 𝜏))
3rspcedvd.4	⊢ (𝜑 → 𝜏)

Assertion

Ref	Expression
3rspcedvd	⊢ (𝜑 → ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 ∃𝑧 ∈ 𝐷 𝜓)

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

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

Proof of Theorem 3rspcedvd

Step	Hyp	Ref	Expression
1		3rspcedvd.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
2		3rspcedvd.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
3	2	2rexbidv 3228	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (∃𝑦 ∈ 𝐷 ∃𝑧 ∈ 𝐷 𝜓 ↔ ∃𝑦 ∈ 𝐷 ∃𝑧 ∈ 𝐷 𝜒))
4		3rspcedvd.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
5		3rspcedvd.2	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (𝜒 ↔ 𝜃))
6	5	rexbidv 3225	. . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (∃𝑧 ∈ 𝐷 𝜒 ↔ ∃𝑧 ∈ 𝐷 𝜃))
7		3rspcedvd.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
8		3rspcedvd.3	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝐶) → (𝜃 ↔ 𝜏))
9		3rspcedvd.4	. . . 4 ⊢ (𝜑 → 𝜏)
10	7, 8, 9	rspcedvd 3555	. . 3 ⊢ (𝜑 → ∃𝑧 ∈ 𝐷 𝜃)
11	4, 6, 10	rspcedvd 3555	. 2 ⊢ (𝜑 → ∃𝑦 ∈ 𝐷 ∃𝑧 ∈ 𝐷 𝜒)
12	1, 3, 11	rspcedvd 3555	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 ∃𝑧 ∈ 𝐷 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∃wrex 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069

This theorem is referenced by: (None)