This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cbveuw

Description: Version of cbveu with a disjoint variable condition, which does not require ax-10 , ax-13 . (Contributed by NM, 25-Nov-1994) (Revised by GG, 23-May-2024)

Ref Expression
Hypotheses cbveuw.1 ⊢ Ⅎ y φ
cbveuw.2 ⊢ Ⅎ x ψ
cbveuw.3 ⊢ x = y → φ ↔ ψ
Assertion cbveuw ⊢ ∃! x φ ↔ ∃! y ψ

Proof

Step Hyp Ref Expression
1 cbveuw.1 ⊢ Ⅎ y φ
2 cbveuw.2 ⊢ Ⅎ x ψ
3 cbveuw.3 ⊢ x = y → φ ↔ ψ
4 1 2 3 cbvexv1 ⊢ ∃ x φ ↔ ∃ y ψ
5 1 2 3 cbvmow ⊢ ∃* x φ ↔ ∃* y ψ
6 4 5 anbi12i ⊢ ∃ x φ ∧ ∃* x φ ↔ ∃ y ψ ∧ ∃* y ψ
7 df-eu ⊢ ∃! x φ ↔ ∃ x φ ∧ ∃* x φ
8 df-eu ⊢ ∃! y ψ ↔ ∃ y ψ ∧ ∃* y ψ
9 6 7 8 3bitr4i ⊢ ∃! x φ ↔ ∃! y ψ