Theorem reutruALT

Description: Alternate proof of reutru . (Contributed by Zhi Wang, 23-Sep-2024) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	reutruALT	$$⊢\exists !xx\in A\leftrightarrow \exists !x\in A\top$$

Proof

Step	Hyp	Ref	Expression
1		rextru	$$⊢\exists xx\in A\leftrightarrow \exists x\in A\top$$
2		rmotru	$$⊢{\exists }^{*}xx\in A\leftrightarrow {\exists }^{*}x\in A\top$$
3	1 2	anbi12i	$$⊢\left(\exists xx\in A\wedge {\exists }^{*}xx\in A\right)\leftrightarrow \left(\exists x\in A\top \wedge {\exists }^{*}x\in A\top \right)$$
4		df-eu	$$⊢\exists !xx\in A\leftrightarrow \left(\exists xx\in A\wedge {\exists }^{*}xx\in A\right)$$
5		reu5	$$⊢\exists !x\in A\top \leftrightarrow \left(\exists x\in A\top \wedge {\exists }^{*}x\in A\top \right)$$
6	3 4 5	3bitr4i	$$⊢\exists !xx\in A\leftrightarrow \exists !x\in A\top$$