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Theorem 3rspcedvd 39384
 Description: Triple application of rspcedvd 3577. (Contributed by Steven Nguyen, 27-Feb-2023.)
Hypotheses
Ref Expression
3rspcedvd.a (𝜑𝐴𝐷)
3rspcedvd.b (𝜑𝐵𝐷)
3rspcedvd.c (𝜑𝐶𝐷)
3rspcedvd.1 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
3rspcedvd.2 ((𝜑𝑦 = 𝐵) → (𝜒𝜃))
3rspcedvd.3 ((𝜑𝑧 = 𝐶) → (𝜃𝜏))
3rspcedvd.4 (𝜑𝜏)
Assertion
Ref Expression
3rspcedvd (𝜑 → ∃𝑥𝐷𝑦𝐷𝑧𝐷 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
StepHypRef Expression
1 3rspcedvd.a . 2 (𝜑𝐴𝐷)
2 3rspcedvd.1 . . 3 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
322rexbidv 3262 . 2 ((𝜑𝑥 = 𝐴) → (∃𝑦𝐷𝑧𝐷 𝜓 ↔ ∃𝑦𝐷𝑧𝐷 𝜒))
4 3rspcedvd.b . . 3 (𝜑𝐵𝐷)
5 3rspcedvd.2 . . . 4 ((𝜑𝑦 = 𝐵) → (𝜒𝜃))
65rexbidv 3259 . . 3 ((𝜑𝑦 = 𝐵) → (∃𝑧𝐷 𝜒 ↔ ∃𝑧𝐷 𝜃))
7 3rspcedvd.c . . . 4 (𝜑𝐶𝐷)
8 3rspcedvd.3 . . . 4 ((𝜑𝑧 = 𝐶) → (𝜃𝜏))
9 3rspcedvd.4 . . . 4 (𝜑𝜏)
107, 8, 9rspcedvd 3577 . . 3 (𝜑 → ∃𝑧𝐷 𝜃)
114, 6, 10rspcedvd 3577 . 2 (𝜑 → ∃𝑦𝐷𝑧𝐷 𝜒)
121, 3, 11rspcedvd 3577 1 (𝜑 → ∃𝑥𝐷𝑦𝐷𝑧𝐷 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2112  ∃wrex 3110 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2794  df-clel 2873  df-ral 3114  df-rex 3115 This theorem is referenced by:  dffltz  39602
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