Metamath Proof Explorer


Theorem cbveu

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 cbveuw , cbveuvw when possible. (Contributed by NM, 25-Nov-1994) (Revised by Mario Carneiro, 7-Oct-2016) (New usage is discouraged.)

Ref Expression
Hypotheses cbveu.1 y φ
cbveu.2 x ψ
cbveu.3 x = y φ ψ
Assertion cbveu ∃! x φ ∃! y ψ

Proof

Step Hyp Ref Expression
1 cbveu.1 y φ
2 cbveu.2 x ψ
3 cbveu.3 x = y φ ψ
4 1 sb8eu ∃! x φ ∃! y y x φ
5 2 3 sbie y x φ ψ
6 5 eubii ∃! y y x φ ∃! y ψ
7 4 6 bitri ∃! x φ ∃! y ψ