reuxfr1dd

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A . Simplifies reuxfr1d . (Contributed by Zhi Wang, 20-Sep-2025)

	Ref	Expression
Hypotheses	reuxfr1dd.1	$⊢ (φ \land y \in C) \to A \in B$
	reuxfr1dd.2	$⊢ (φ \land x \in B) \to \exists! y \in C x = A$
	reuxfr1dd.3	$⊢ (φ \land (y \in C \land x = A)) \to (ψ \leftrightarrow χ)$
Assertion	reuxfr1dd	$⊢ φ \to (\exists! x \in B ψ \leftrightarrow \exists! y \in C χ)$

Proof

Step	Hyp	Ref	Expression
1		reuxfr1dd.1	$⊢ (φ \land y \in C) \to A \in B$
2		reuxfr1dd.2	$⊢ (φ \land x \in B) \to \exists! y \in C x = A$
3		reuxfr1dd.3	$⊢ (φ \land (y \in C \land x = A)) \to (ψ \leftrightarrow χ)$
4		reurex	$⊢ \exists! y \in C x = A \to \exists y \in C x = A$
5	2 4	syl	$⊢ (φ \land x \in B) \to \exists y \in C x = A$
6	5	biantrurd	$⊢ (φ \land x \in B) \to (ψ \leftrightarrow (\exists y \in C x = A \land ψ))$
7		r19.41v	$⊢ \exists y \in C (x = A \land ψ) \leftrightarrow (\exists y \in C x = A \land ψ)$
8	3	pm5.32da	$⊢ φ \to (((y \in C \land x = A) \land ψ) \leftrightarrow ((y \in C \land x = A) \land χ))$
9		anass	$⊢ ((y \in C \land x = A) \land ψ) \leftrightarrow (y \in C \land (x = A \land ψ))$
10		anass	$⊢ ((y \in C \land x = A) \land χ) \leftrightarrow (y \in C \land (x = A \land χ))$
11	8 9 10	3bitr3g	$⊢ φ \to ((y \in C \land (x = A \land ψ)) \leftrightarrow (y \in C \land (x = A \land χ)))$
12	11	rexbidv2	$⊢ φ \to (\exists y \in C (x = A \land ψ) \leftrightarrow \exists y \in C (x = A \land χ))$
13	7 12	bitr3id	$⊢ φ \to ((\exists y \in C x = A \land ψ) \leftrightarrow \exists y \in C (x = A \land χ))$
14	13	adantr	$⊢ (φ \land x \in B) \to ((\exists y \in C x = A \land ψ) \leftrightarrow \exists y \in C (x = A \land χ))$
15	6 14	bitrd	$⊢ (φ \land x \in B) \to (ψ \leftrightarrow \exists y \in C (x = A \land χ))$
16	15	reubidva	$⊢ φ \to (\exists! x \in B ψ \leftrightarrow \exists! x \in B \exists y \in C (x = A \land χ))$
17		reurmo	$⊢ \exists! y \in C x = A \to \exists^{*} y \in C x = A$
18	2 17	syl	$⊢ (φ \land x \in B) \to \exists^{*} y \in C x = A$
19	1 18	reuxfrd	$⊢ φ \to (\exists! x \in B \exists y \in C (x = A \land χ) \leftrightarrow \exists! y \in C χ)$
20	16 19	bitrd	$⊢ φ \to (\exists! x \in B ψ \leftrightarrow \exists! y \in C χ)$

Metamath Proof Explorer

Theorem reuxfr1dd

Proof