reuxfrd

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A . (Contributed by NM, 16-Jan-2012) Separate variables B and C. (Revised by Thierry Arnoux, 8-Oct-2017)

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

Proof

Step	Hyp	Ref	Expression
1		reuxfrd.1	$⊢ (φ \land y \in C) \to A \in B$
2		reuxfrd.2	$⊢ (φ \land x \in B) \to \exists^{*} y \in C x = A$
3		rmoan	$⊢ \exists^{} y \in C x = A \to \exists^{} y \in C (ψ \land x = A)$
4	2 3	syl	$⊢ (φ \land x \in B) \to \exists^{*} y \in C (ψ \land x = A)$
5		ancom	$⊢ (ψ \land x = A) \leftrightarrow (x = A \land ψ)$
6	5	rmobii	$⊢ \exists^{} y \in C (ψ \land x = A) \leftrightarrow \exists^{} y \in C (x = A \land ψ)$
7	4 6	sylib	$⊢ (φ \land x \in B) \to \exists^{*} y \in C (x = A \land ψ)$
8	7	ralrimiva	$⊢ φ \to \forall x \in B \exists^{*} y \in C (x = A \land ψ)$
9		2reuswap	$⊢ \forall x \in B \exists^{*} y \in C (x = A \land ψ) \to (\exists! x \in B \exists y \in C (x = A \land ψ) \to \exists! y \in C \exists x \in B (x = A \land ψ))$
10	8 9	syl	$⊢ φ \to (\exists! x \in B \exists y \in C (x = A \land ψ) \to \exists! y \in C \exists x \in B (x = A \land ψ))$
11		2reuswap2	$⊢ \forall y \in C \exists^{*} x (x \in B \land (x = A \land ψ)) \to (\exists! y \in C \exists x \in B (x = A \land ψ) \to \exists! x \in B \exists y \in C (x = A \land ψ))$
12		moeq	$⊢ \exists^{*} x x = A$
13	12	moani	$⊢ \exists^{*} x ((x \in B \land ψ) \land x = A)$
14		ancom	$⊢ ((x \in B \land ψ) \land x = A) \leftrightarrow (x = A \land (x \in B \land ψ))$
15		an12	$⊢ (x = A \land (x \in B \land ψ)) \leftrightarrow (x \in B \land (x = A \land ψ))$
16	14 15	bitri	$⊢ ((x \in B \land ψ) \land x = A) \leftrightarrow (x \in B \land (x = A \land ψ))$
17	16	mobii	$⊢ \exists^{} x ((x \in B \land ψ) \land x = A) \leftrightarrow \exists^{} x (x \in B \land (x = A \land ψ))$
18	13 17	mpbi	$⊢ \exists^{*} x (x \in B \land (x = A \land ψ))$
19	18	a1i	$⊢ y \in C \to \exists^{*} x (x \in B \land (x = A \land ψ))$
20	11 19	mprg	$⊢ \exists! y \in C \exists x \in B (x = A \land ψ) \to \exists! x \in B \exists y \in C (x = A \land ψ)$
21	10 20	impbid1	$⊢ φ \to (\exists! x \in B \exists y \in C (x = A \land ψ) \leftrightarrow \exists! y \in C \exists x \in B (x = A \land ψ))$
22		biidd	$⊢ x = A \to (ψ \leftrightarrow ψ)$
23	22	ceqsrexv	$⊢ A \in B \to (\exists x \in B (x = A \land ψ) \leftrightarrow ψ)$
24	1 23	syl	$⊢ (φ \land y \in C) \to (\exists x \in B (x = A \land ψ) \leftrightarrow ψ)$
25	24	reubidva	$⊢ φ \to (\exists! y \in C \exists x \in B (x = A \land ψ) \leftrightarrow \exists! y \in C ψ)$
26	21 25	bitrd	$⊢ φ \to (\exists! x \in B \exists y \in C (x = A \land ψ) \leftrightarrow \exists! y \in C ψ)$

Metamath Proof Explorer

Theorem reuxfrd

Proof