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Metamath Proof Explorer

Theorem cbv3h

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbv3hv if possible. (Contributed by NM, 8-Jun-1993) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 12-May-2018) (New usage is discouraged.)

Ref Expression
Hypotheses cbv3h.1 ⊢ φ → ∀ y φ
cbv3h.2 ⊢ ψ → ∀ x ψ
cbv3h.3 ⊢ x = y → φ → ψ
Assertion cbv3h ⊢ ∀ x φ → ∀ y ψ

Proof

Step Hyp Ref Expression
1 cbv3h.1 ⊢ φ → ∀ y φ
2 cbv3h.2 ⊢ ψ → ∀ x ψ
3 cbv3h.3 ⊢ x = y → φ → ψ
4 1 nf5i ⊢ Ⅎ y φ
5 2 nf5i ⊢ Ⅎ x ψ
6 4 5 3 cbv3 ⊢ ∀ x φ → ∀ y ψ