Theorem cbvral3vw 3251

Description: Change bound variables of triple restricted universal quantification, using implicit substitution. Version of cbvral3v 3361 with a disjoint variable condition, which does not require ax-13 2406. (Contributed by NM, 10-May-2005.) Avoid ax-13 2406. (Revised by GG, 10-Jan-2024.)

Hypotheses

Ref	Expression
cbvral3vw.1	⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒))
cbvral3vw.2	⊢ (𝑦 = 𝑣 → (𝜒 ↔ 𝜃))
cbvral3vw.3	⊢ (𝑧 = 𝑢 → (𝜃 ↔ 𝜓))

Assertion

Ref	Expression
cbvral3vw	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓)

Distinct variable groups:   𝜑,𝑤   𝜓,𝑧   𝜒,𝑥   𝜒,𝑣   𝜃,𝑦   𝜃,𝑢   𝑥,𝐴   𝑧,𝑢   𝑤,𝐴   𝑥,𝑦,𝐵,𝑤   𝑣,𝐵   𝑥,𝑧,𝐶,𝑦,𝑤   𝑧,𝑣,𝐶   𝑢,𝐶   𝑦,𝑣

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧, 𝑣, 𝑢)   𝜓(𝑥, 𝑦, 𝑤, 𝑣, 𝑢)   𝜒(𝑦, 𝑧, 𝑤, 𝑢)   𝜃(𝑥, 𝑧, 𝑤, 𝑣)   𝐴(𝑦, 𝑧, 𝑣, 𝑢)   𝐵(𝑧, 𝑢)

Proof of Theorem cbvral3vw

Step	Hyp	Ref	Expression
1		cbvral3vw.1	. . . 4 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒))
2	1	2ralbidv 3231	. . 3 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒))
3	2	cbvralvw 3245	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒)
4		cbvral3vw.2	. . . 4 ⊢ (𝑦 = 𝑣 → (𝜒 ↔ 𝜃))
5		cbvral3vw.3	. . . 4 ⊢ (𝑧 = 𝑢 → (𝜃 ↔ 𝜓))
6	4, 5	cbvral2vw 3249	. . 3 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒 ↔ ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓)
7	6	ralbii 3113	. 2 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒 ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓)
8	3, 7	bitri 278	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wral 3081

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-clel 2840  df-ral 3082

This theorem is used by:  cbvral4vw  3252  latdisd  18579  addsprop  28224  dffltz  43443  isthincd2lem2  50289