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Metamath Proof Explorer

Theorem reuxfr

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A . (Contributed by NM, 14-Nov-2004) (Revised by NM, 16-Jun-2017)

	Ref	Expression
Hypotheses	reuxfr.1	$⊢ y \in C \to A \in B$
	reuxfr.2	$⊢ x \in B \to \exists^{*} y \in C x = A$
Assertion	reuxfr	$⊢ \exists! x \in B \exists y \in C (x = A \land φ) \leftrightarrow \exists! y \in C φ$

Proof

Step	Hyp	Ref	Expression
1		reuxfr.1	$⊢ y \in C \to A \in B$
2		reuxfr.2	$⊢ x \in B \to \exists^{*} y \in C x = A$
3	1	adantl	$⊢ (⊤ \land y \in C) \to A \in B$
4	2	adantl	$⊢ (⊤ \land x \in B) \to \exists^{*} y \in C x = A$
5	3 4	reuxfrd	$⊢ ⊤ \to (\exists! x \in B \exists y \in C (x = A \land φ) \leftrightarrow \exists! y \in C φ)$
6	5	mptru	$⊢ \exists! x \in B \exists y \in C (x = A \land φ) \leftrightarrow \exists! y \in C φ$