Theorem iotacl

Description: Membership law for descriptions.

This can be useful for expanding an unbounded iota-based definition (see df-iota ). If you have a bounded iota-based definition, riotacl2 may be useful.

(Contributed by Andrew Salmon, 1-Aug-2011)

	Ref	Expression
Assertion	iotacl	$$⊢\exists !x\phi \to \left(\iota x|\phi \right)\in \left\{x|\phi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		iota4	$$⊢\exists !x\phi \to \underset{\dot{}}{[}\left(\iota x|\phi \right)/x\underset{\dot{}}{]}\phi$$
2		df-sbc	$$⊢\underset{\dot{}}{[}\left(\iota x|\phi \right)/x\underset{\dot{}}{]}\phi \leftrightarrow \left(\iota x|\phi \right)\in \left\{x|\phi \right\}$$
3	1 2	sylib	$$⊢\exists !x\phi \to \left(\iota x|\phi \right)\in \left\{x|\phi \right\}$$