eubii

Description: Introduce unique existential quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994) (Revised by Mario Carneiro, 6-Oct-2016) Avoid ax-5 . (Revised by Wolf Lammen, 27-Sep-2023)

		Ref	Expression
	Hypothesis	eubii.1	$⊢ φ \leftrightarrow ψ$
	Assertion	eubii	$⊢ \exists! x φ \leftrightarrow \exists! x ψ$

Proof

Step	Hyp	Ref	Expression
1		eubii.1	$⊢ φ \leftrightarrow ψ$
2	1	exbii	$⊢ \exists x φ \leftrightarrow \exists x ψ$
3	1	mobii	$⊢ \exists^{} x φ \leftrightarrow \exists^{} x ψ$
4	2 3	anbi12i	$⊢ (\exists x φ \land \exists^{} x φ) \leftrightarrow (\exists x ψ \land \exists^{} x ψ)$
5		df-eu	$⊢ \exists! x φ \leftrightarrow (\exists x φ \land \exists^{*} x φ)$
6		df-eu	$⊢ \exists! x ψ \leftrightarrow (\exists x ψ \land \exists^{*} x ψ)$
7	4 5 6	3bitr4i	$⊢ \exists! x φ \leftrightarrow \exists! x ψ$

Metamath Proof Explorer

Theorem eubii

Proof