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

Metamath Proof Explorer

Theorem eubid

Description: Formula-building rule for the unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994) (Proof shortened by Wolf Lammen, 19-Feb-2023)

	Ref	Expression
Hypotheses	eubid.1	$⊢ Ⅎ x φ$
	eubid.2	$⊢ φ \to (ψ \leftrightarrow χ)$
Assertion	eubid	$⊢ φ \to (\exists! x ψ \leftrightarrow \exists! x χ)$

Proof

Step	Hyp	Ref	Expression
1		eubid.1	$⊢ Ⅎ x φ$
2		eubid.2	$⊢ φ \to (ψ \leftrightarrow χ)$
3	1 2	alrimi	$⊢ φ \to \forall x (ψ \leftrightarrow χ)$
4		eubi	$⊢ \forall x (ψ \leftrightarrow χ) \to (\exists! x ψ \leftrightarrow \exists! x χ)$
5	3 4	syl	$⊢ φ \to (\exists! x ψ \leftrightarrow \exists! x χ)$