cbvreuw

Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu with a disjoint variable condition, which does not require ax-13 . (Contributed by Mario Carneiro, 15-Oct-2016) Avoid ax-13 . (Revised by GG, 10-Jan-2024) Avoid ax-10 . (Revised by Wolf Lammen, 10-Dec-2024)

	Ref	Expression
Hypotheses	cbvreuw.1	$⊢ Ⅎ y φ$
	cbvreuw.2	$⊢ Ⅎ x ψ$
	cbvreuw.3	$⊢ x = y \to (φ \leftrightarrow ψ)$
Assertion	cbvreuw	$⊢ \exists! x \in A φ \leftrightarrow \exists! y \in A ψ$

Proof

Step	Hyp	Ref	Expression
1		cbvreuw.1	$⊢ Ⅎ y φ$
2		cbvreuw.2	$⊢ Ⅎ x ψ$
3		cbvreuw.3	$⊢ x = y \to (φ \leftrightarrow ψ)$
4	1 2 3	cbvrexw	$⊢ \exists x \in A φ \leftrightarrow \exists y \in A ψ$
5	1 2 3	cbvrmow	$⊢ \exists^{} x \in A φ \leftrightarrow \exists^{} y \in A ψ$
6	4 5	anbi12i	$⊢ (\exists x \in A φ \land \exists^{} x \in A φ) \leftrightarrow (\exists y \in A ψ \land \exists^{} y \in A ψ)$
7		reu5	$⊢ \exists! x \in A φ \leftrightarrow (\exists x \in A φ \land \exists^{*} x \in A φ)$
8		reu5	$⊢ \exists! y \in A ψ \leftrightarrow (\exists y \in A ψ \land \exists^{*} y \in A ψ)$
9	6 7 8	3bitr4i	$⊢ \exists! x \in A φ \leftrightarrow \exists! y \in A ψ$

Metamath Proof Explorer

Theorem cbvreuw

Proof