Theorem 3rspcedvd 38034

Description: Triple application of rspcedvd 3533. (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 3267	. 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 3262	. . 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 3533	. . 3 ⊢ (𝜑 → ∃𝑧 ∈ 𝐷 𝜃)
11	4, 6, 10	rspcedvd 3533	. 2 ⊢ (𝜑 → ∃𝑦 ∈ 𝐷 ∃𝑧 ∈ 𝐷 𝜒)
12	1, 3, 11	rspcedvd 3533	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 ∃𝑧 ∈ 𝐷 𝜓)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164  ∃wrex 3118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-v 3416

This theorem is referenced by:  dffltz  38092