Metamath Proof Explorer

Theorem reuxfr1d

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A . Cf. reuxfr1ds . (Contributed by Thierry Arnoux, 7-Apr-2017)

		Ref	Expression
	Hypotheses	reuxfr1d.1	$⊢ (φ \land y \in C) \to A \in B$
		reuxfr1d.2	$⊢ (φ \land x \in B) \to \exists! y \in C x = A$
		reuxfr1d.3	$⊢ (φ \land x = A) \to (ψ \leftrightarrow χ)$
	Assertion	reuxfr1d	$⊢ φ \to (\exists! x \in B ψ \leftrightarrow \exists! y \in C χ)$

Proof

Step	Hyp	Ref	Expression
1		reuxfr1d.1	$⊢ (φ \land y \in C) \to A \in B$
2		reuxfr1d.2	$⊢ (φ \land x \in B) \to \exists! y \in C x = A$
3		reuxfr1d.3	$⊢ (φ \land x = A) \to (ψ \leftrightarrow χ)$
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	$⊢ φ \to ((x = A \land ψ) \leftrightarrow (x = A \land χ))$
9	8	rexbidv	$⊢ φ \to (\exists y \in C (x = A \land ψ) \leftrightarrow \exists y \in C (x = A \land χ))$
10	7 9	bitr3id	$⊢ φ \to ((\exists y \in C x = A \land ψ) \leftrightarrow \exists y \in C (x = A \land χ))$
11	10	adantr	$⊢ (φ \land x \in B) \to ((\exists y \in C x = A \land ψ) \leftrightarrow \exists y \in C (x = A \land χ))$
12	6 11	bitrd	$⊢ (φ \land x \in B) \to (ψ \leftrightarrow \exists y \in C (x = A \land χ))$
13	12	reubidva	$⊢ φ \to (\exists! x \in B ψ \leftrightarrow \exists! x \in B \exists y \in C (x = A \land χ))$
14		reurmo	$⊢ \exists! y \in C x = A \to \exists^{*} y \in C x = A$
15	2 14	syl	$⊢ (φ \land x \in B) \to \exists^{*} y \in C x = A$
16	1 15	reuxfrd	$⊢ φ \to (\exists! x \in B \exists y \in C (x = A \land χ) \leftrightarrow \exists! y \in C χ)$
17	13 16	bitrd	$⊢ φ \to (\exists! x \in B ψ \leftrightarrow \exists! y \in C χ)$