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Metamath Proof Explorer

Theorem mobid

Description: Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by NM, 8-Mar-1995) Remove dependency on ax-10 , ax-11 , ax-13 . (Revised by BJ, 14-Oct-2022) (Proof shortened by Wolf Lammen, 18-Feb-2023)

Ref Expression
Hypotheses mobid.1 ⊢ Ⅎ x φ
mobid.2 ⊢ φ → ψ ↔ χ
Assertion mobid ⊢ φ → ∃* x ψ ↔ ∃* x χ

Proof

Step Hyp Ref Expression
1 mobid.1 ⊢ Ⅎ x φ
2 mobid.2 ⊢ φ → ψ ↔ χ
3 1 2 alrimi ⊢ φ → ∀ x ψ ↔ χ
4 mobi ⊢ ∀ x ψ ↔ χ → ∃* x ψ ↔ ∃* x χ
5 3 4 syl ⊢ φ → ∃* x ψ ↔ ∃* x χ