Theorem cbvral6vw 3241

Description: Change bound variables of sextuple restricted universal quantification, using implicit substitution. (Contributed by Scott Fenton, 5-Mar-2025.)

Hypotheses

Ref	Expression
cbvral6vw.1	⊢ (𝑥 = 𝑎 → (𝜑 ↔ 𝜒))
cbvral6vw.2	⊢ (𝑦 = 𝑏 → (𝜒 ↔ 𝜃))
cbvral6vw.3	⊢ (𝑧 = 𝑐 → (𝜃 ↔ 𝜏))
cbvral6vw.4	⊢ (𝑤 = 𝑑 → (𝜏 ↔ 𝜂))
cbvral6vw.5	⊢ (𝑝 = 𝑒 → (𝜂 ↔ 𝜁))
cbvral6vw.6	⊢ (𝑞 = 𝑓 → (𝜁 ↔ 𝜓))

Assertion

Ref	Expression
cbvral6vw	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜑 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐶 ∀𝑑 ∈ 𝐷 ∀𝑒 ∈ 𝐸 ∀𝑓 ∈ 𝐹 𝜓)

Distinct variable groups:   𝑥,𝑎,𝐴   𝑥,𝑦,𝐵,𝑎   𝑦,𝑏,𝐵   𝑥,𝐶,𝑦,𝑎   𝐶,𝑏   𝑧,𝑐,𝐶   𝑥,𝑧,𝐷,𝑦,𝑎   𝐷,𝑏   𝐷,𝑐   𝑤,𝑑,𝐷   𝑥,𝐸,𝑦,𝑧,𝑎   𝐸,𝑏   𝐸,𝑐   𝐸,𝑑   𝑒,𝐸   𝑤,𝑝,𝐸   𝑥,𝐹,𝑦,𝑧,𝑎   𝐹,𝑏   𝐹,𝑐   𝐹,𝑑   𝑒,𝐹   𝑓,𝑞,𝐹   𝐹,𝑝,𝑤,𝑎   𝜑,𝑎   𝑞,𝑎,𝑤   𝜒,𝑏   𝑝,𝑏,𝑞,𝑧,𝑤   𝑝,𝑐,𝑞   𝜃,𝑐   𝑤,𝑐   𝜒,𝑥   𝑝,𝑑,𝑞   𝜏,𝑑   𝜂,𝑒   𝑒,𝑝,𝑞   𝜂,𝑤   𝜁,𝑓   𝑥,𝑞,𝑦,𝑝   𝜁,𝑝   𝜓,𝑞   𝑥,𝑤,𝑦   𝜏,𝑧   𝜃,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑒,𝑓,𝑞,𝑝,𝑏,𝑐,𝑑)   𝜓(𝑥,𝑦,𝑧,𝑤,𝑒,𝑓,𝑝,𝑎,𝑏,𝑐,𝑑)   𝜒(𝑦,𝑧,𝑤,𝑒,𝑓,𝑞,𝑝,𝑎,𝑐,𝑑)   𝜃(𝑥,𝑧,𝑤,𝑒,𝑓,𝑞,𝑝,𝑎,𝑏,𝑑)   𝜏(𝑥,𝑦,𝑤,𝑒,𝑓,𝑞,𝑝,𝑎,𝑏,𝑐)   𝜂(𝑥,𝑦,𝑧,𝑓,𝑞,𝑝,𝑎,𝑏,𝑐,𝑑)   𝜁(𝑥,𝑦,𝑧,𝑤,𝑒,𝑞,𝑎,𝑏,𝑐,𝑑)   𝐴(𝑦,𝑧,𝑤,𝑒,𝑓,𝑞,𝑝,𝑏,𝑐,𝑑)   𝐵(𝑧,𝑤,𝑒,𝑓,𝑞,𝑝,𝑐,𝑑)   𝐶(𝑤,𝑒,𝑓,𝑞,𝑝,𝑑)   𝐷(𝑒,𝑓,𝑞,𝑝)   𝐸(𝑓,𝑞)

Proof of Theorem cbvral6vw

Step	Hyp	Ref	Expression
1		cbvral6vw.1	. . . 4 ⊢ (𝑥 = 𝑎 → (𝜑 ↔ 𝜒))
2	1	2ralbidv 3217	. . 3 ⊢ (𝑥 = 𝑎 → (∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜑 ↔ ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜒))
3		cbvral6vw.2	. . . 4 ⊢ (𝑦 = 𝑏 → (𝜒 ↔ 𝜃))
4	3	2ralbidv 3217	. . 3 ⊢ (𝑦 = 𝑏 → (∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜒 ↔ ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜃))
5		cbvral6vw.3	. . . 4 ⊢ (𝑧 = 𝑐 → (𝜃 ↔ 𝜏))
6	5	2ralbidv 3217	. . 3 ⊢ (𝑧 = 𝑐 → (∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜃 ↔ ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜏))
7		cbvral6vw.4	. . . 4 ⊢ (𝑤 = 𝑑 → (𝜏 ↔ 𝜂))
8	7	2ralbidv 3217	. . 3 ⊢ (𝑤 = 𝑑 → (∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜏 ↔ ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜂))
9	2, 4, 6, 8	cbvral4vw 3240	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜑 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐶 ∀𝑑 ∈ 𝐷 ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜂)
10		cbvral6vw.5	. . . 4 ⊢ (𝑝 = 𝑒 → (𝜂 ↔ 𝜁))
11		cbvral6vw.6	. . . 4 ⊢ (𝑞 = 𝑓 → (𝜁 ↔ 𝜓))
12	10, 11	cbvral2vw 3237	. . 3 ⊢ (∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜂 ↔ ∀𝑒 ∈ 𝐸 ∀𝑓 ∈ 𝐹 𝜓)
13	12	4ralbii 3130	. 2 ⊢ (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐶 ∀𝑑 ∈ 𝐷 ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜂 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐶 ∀𝑑 ∈ 𝐷 ∀𝑒 ∈ 𝐸 ∀𝑓 ∈ 𝐹 𝜓)
14	9, 13	bitri 274	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜑 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐶 ∀𝑑 ∈ 𝐷 ∀𝑒 ∈ 𝐸 ∀𝑓 ∈ 𝐹 𝜓)

Syntax hints:   → wi 4   ↔ wb 205  ∀wral 3060

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 1913  ax-6 1971  ax-7 2011  ax-8 2108

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-clel 2809  df-ral 3061

This theorem is referenced by:  mulsproplemcbv  27484  mulsprop  27499