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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 ⊢ Ⅎ y φ
cbveu.2 ⊢ Ⅎ x ψ
cbveu.3 ⊢ x = y → φ ↔ ψ
Assertion cbveu ⊢ ∃! x φ ↔ ∃! y ψ

Proof

Step Hyp Ref Expression
1 cbveu.1 ⊢ Ⅎ y φ
2 cbveu.2 ⊢ Ⅎ 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