Theorem dfaiota3

Description: Alternate definition of iota' , using the if operator: this is to df-aiota what dfiota4 is to df-iota . It is simpler than df-aiota and uses no dummy variables, so it would be the preferred definition if iota' becomes the description binder used in set.mm. (Contributed by BJ, 31-Aug-2024)

	Ref	Expression
Assertion	dfaiota3	$$⊢\iota =if\left(\exists !x\phi ,\bigcap \left\{x|\phi \right\},\mathrm{V}\right)$$

Proof

Step	Hyp	Ref	Expression
1		aiotaint	$$⊢\exists !x\phi \to \iota =\bigcap \left\{x|\phi \right\}$$
2		aiotavb	$$⊢\neg \exists !x\phi \leftrightarrow \iota =\mathrm{V}$$
3	2	biimpi	$$⊢\neg \exists !x\phi \to \iota =\mathrm{V}$$
4		ifval	$$⊢\iota =if\left(\exists !x\phi ,\bigcap \left\{x|\phi \right\},\mathrm{V}\right)\leftrightarrow \left(\left(\exists !x\phi \to \iota =\bigcap \left\{x|\phi \right\}\right)\wedge \left(\neg \exists !x\phi \to \iota =\mathrm{V}\right)\right)$$
5	1 3 4	mpbir2an	$$⊢\iota =if\left(\exists !x\phi ,\bigcap \left\{x|\phi \right\},\mathrm{V}\right)$$