Theorem moelOLD

Description: Obsolete version of moel as of 23-Nov-2024. (Contributed by Thierry Arnoux, 26-Jul-2018) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	moelOLD	$$⊢{\exists }^{*}xx\in A\leftrightarrow \forall x\in A\forall y\in Ax=y$$

Proof

Step	Hyp	Ref	Expression
1		ralcom4	$$⊢\forall x\in A\forall y\left(y\in A\to x=y\right)\leftrightarrow \forall y\forall x\in A\left(y\in A\to x=y\right)$$
2		df-ral	$$⊢\forall y\in Ax=y\leftrightarrow \forall y\left(y\in A\to x=y\right)$$
3	2	ralbii	$$⊢\forall x\in A\forall y\in Ax=y\leftrightarrow \forall x\in A\forall y\left(y\in A\to x=y\right)$$
4		alcom	$$⊢\forall x\forall y\left(\left(x\in A\wedge y\in A\right)\to x=y\right)\leftrightarrow \forall y\forall x\left(\left(x\in A\wedge y\in A\right)\to x=y\right)$$
5		eleq1w	$$⊢x=y\to \left(x\in A\leftrightarrow y\in A\right)$$
6	5	mo4	$$⊢{\exists }^{*}xx\in A\leftrightarrow \forall x\forall y\left(\left(x\in A\wedge y\in A\right)\to x=y\right)$$
7		df-ral	$$⊢\forall x\in A\left(y\in A\to x=y\right)\leftrightarrow \forall x\left(x\in A\to \left(y\in A\to x=y\right)\right)$$
8		impexp	$$⊢\left(\left(x\in A\wedge y\in A\right)\to x=y\right)\leftrightarrow \left(x\in A\to \left(y\in A\to x=y\right)\right)$$
9	8	albii	$$⊢\forall x\left(\left(x\in A\wedge y\in A\right)\to x=y\right)\leftrightarrow \forall x\left(x\in A\to \left(y\in A\to x=y\right)\right)$$
10	7 9	bitr4i	$$⊢\forall x\in A\left(y\in A\to x=y\right)\leftrightarrow \forall x\left(\left(x\in A\wedge y\in A\right)\to x=y\right)$$
11	10	albii	$$⊢\forall y\forall x\in A\left(y\in A\to x=y\right)\leftrightarrow \forall y\forall x\left(\left(x\in A\wedge y\in A\right)\to x=y\right)$$
12	4 6 11	3bitr4i	$$⊢{\exists }^{*}xx\in A\leftrightarrow \forall y\forall x\in A\left(y\in A\to x=y\right)$$
13	1 3 12	3bitr4ri	$$⊢{\exists }^{*}xx\in A\leftrightarrow \forall x\in A\forall y\in Ax=y$$