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Metamath Proof Explorer

Theorem cbv3hv

Description: Rule used to change bound variables, using implicit substitution. Version of cbv3h with a disjoint variable condition on x , y , which does not require ax-13 . Was used in a proof of axc11n (but of independent interest). (Contributed by NM, 25-Jul-2015) (Proof shortened by Wolf Lammen, 29-Nov-2020) (Proof shortened by BJ, 30-Nov-2020)

Ref Expression
Hypotheses cbv3hv.nf1 ⊢ φ → ∀ y φ
cbv3hv.nf2 ⊢ ψ → ∀ x ψ
cbv3hv.1 ⊢ x = y → φ → ψ
Assertion cbv3hv ⊢ ∀ x φ → ∀ y ψ

Proof

Step Hyp Ref Expression
1 cbv3hv.nf1 ⊢ φ → ∀ y φ
2 cbv3hv.nf2 ⊢ ψ → ∀ x ψ
3 cbv3hv.1 ⊢ x = y → φ → ψ
4 1 nf5i ⊢ Ⅎ y φ
5 2 nf5i ⊢ Ⅎ x ψ
6 4 5 3 cbv3v ⊢ ∀ x φ → ∀ y ψ