Metamath Proof Explorer


Theorem cbvrmo

Description: Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvrmow , cbvrmovw when possible. (Contributed by NM, 16-Jun-2017) (New usage is discouraged.)

Ref Expression
Hypotheses cbvral.1 y φ
cbvral.2 x ψ
cbvral.3 x = y φ ψ
Assertion cbvrmo * x A φ * y A ψ

Proof

Step Hyp Ref Expression
1 cbvral.1 y φ
2 cbvral.2 x ψ
3 cbvral.3 x = y φ ψ
4 1 2 3 cbvrex x A φ y A ψ
5 1 2 3 cbvreu ∃! x A φ ∃! y A ψ
6 4 5 imbi12i x A φ ∃! x A φ y A ψ ∃! y A ψ
7 rmo5 * x A φ x A φ ∃! x A φ
8 rmo5 * y A ψ y A ψ ∃! y A ψ
9 6 7 8 3bitr4i * x A φ * y A ψ