Theorem wl-dfrmof

Description: Restricted "at most one" ( df-wl-rmo ) allows a simplification, if we can assume all occurrences of x in A are bounded. (Contributed by Wolf Lammen, 28-May-2023)

	Ref	Expression
Assertion	wl-dfrmof	$$⊢\underset{\_}{Ⅎ}xA\to \left({\exists }^{*}x:A\phi \leftrightarrow {\exists }^{*}x\left(x\in A\wedge \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		wl-dfralf	$$⊢\underset{\_}{Ⅎ}xA\to \left(\forall x:A\left(\phi \to x=y\right)\leftrightarrow \forall x\left(x\in A\to \left(\phi \to x=y\right)\right)\right)$$
2		nfnfc1	$$⊢Ⅎx\underset{\_}{Ⅎ}xA$$
3		impexp	$$⊢\left(\left(x\in A\wedge \phi \right)\to x=y\right)\leftrightarrow \left(x\in A\to \left(\phi \to x=y\right)\right)$$
4	3	a1i	$$⊢\underset{\_}{Ⅎ}xA\to \left(\left(\left(x\in A\wedge \phi \right)\to x=y\right)\leftrightarrow \left(x\in A\to \left(\phi \to x=y\right)\right)\right)$$
5	2 4	albid	$$⊢\underset{\_}{Ⅎ}xA\to \left(\forall x\left(\left(x\in A\wedge \phi \right)\to x=y\right)\leftrightarrow \forall x\left(x\in A\to \left(\phi \to x=y\right)\right)\right)$$
6	1 5	bitr4d	$$⊢\underset{\_}{Ⅎ}xA\to \left(\forall x:A\left(\phi \to x=y\right)\leftrightarrow \forall x\left(\left(x\in A\wedge \phi \right)\to x=y\right)\right)$$
7	6	exbidv	$$⊢\underset{\_}{Ⅎ}xA\to \left(\exists y\forall x:A\left(\phi \to x=y\right)\leftrightarrow \exists y\forall x\left(\left(x\in A\wedge \phi \right)\to x=y\right)\right)$$
8		df-wl-rmo	$$⊢{\exists }^{*}x:A\phi \leftrightarrow \exists y\forall x:A\left(\phi \to x=y\right)$$
9		df-mo	$$⊢{\exists }^{*}x\left(x\in A\wedge \phi \right)\leftrightarrow \exists y\forall x\left(\left(x\in A\wedge \phi \right)\to x=y\right)$$
10	7 8 9	3bitr4g	$$⊢\underset{\_}{Ⅎ}xA\to \left({\exists }^{*}x:A\phi \leftrightarrow {\exists }^{*}x\left(x\in A\wedge \phi \right)\right)$$