Metamath Proof Explorer

Theorem cbvral3v

Description: Change bound variables of triple restricted universal quantification, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvral3vw when possible. (Contributed by NM, 10-May-2005) (New usage is discouraged.)

Ref Expression
Hypotheses cbvral3v.1 ⊢x=w→φ↔χ
cbvral3v.2 ⊢y=v→χ↔θ
cbvral3v.3 ⊢z=u→θ↔ψ
Assertion cbvral3v ⊢∀x∈A∀y∈B∀z∈Cφ↔∀w∈A∀v∈B∀u∈Cψ

Proof

Step Hyp Ref Expression
1 cbvral3v.1 ⊢x=w→φ↔χ
2 cbvral3v.2 ⊢y=v→χ↔θ
3 cbvral3v.3 ⊢z=u→θ↔ψ
4 1 2ralbidv ⊢x=w→∀y∈B∀z∈Cφ↔∀y∈B∀z∈Cχ
5 4 cbvralv ⊢∀x∈A∀y∈B∀z∈Cφ↔∀w∈A∀y∈B∀z∈Cχ
6 2 3 cbvral2v ⊢∀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