Metamath Proof Explorer

Theorem cbvral3vw

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

Ref Expression
Hypotheses cbvral3vw.1 ⊢x=w→φ↔χ
cbvral3vw.2 ⊢y=v→χ↔θ
cbvral3vw.3 ⊢z=u→θ↔ψ
Assertion cbvral3vw ⊢∀x∈A∀y∈B∀z∈Cφ↔∀w∈A∀v∈B∀u∈Cψ

Proof

Step Hyp Ref Expression
1 cbvral3vw.1 ⊢x=w→φ↔χ
2 cbvral3vw.2 ⊢y=v→χ↔θ
3 cbvral3vw.3 ⊢z=u→θ↔ψ
4 1 2ralbidv ⊢x=w→∀y∈B∀z∈Cφ↔∀y∈B∀z∈Cχ
5 4 cbvralvw ⊢∀x∈A∀y∈B∀z∈Cφ↔∀w∈A∀y∈B∀z∈Cχ
6 2 3 cbvral2vw ⊢∀y∈B∀z∈Cχ↔∀v∈B∀u∈Cψ
7 6 ralbii ⊢∀w∈A∀y∈B∀z∈Cχ↔∀w∈A∀v∈B∀u∈Cψ
8 5 7 bitri ⊢∀x∈A∀y∈B∀z∈Cφ↔∀w∈A∀v∈B∀u∈Cψ