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	⊢ ( ∃! 𝑥 𝜑 → ( ℩ 𝑥 𝜑 ) ∈ { 𝑥 ∣ 𝜑 } )

Proof

Step	Hyp	Ref	Expression
1		iota4	⊢ ( ∃! 𝑥 𝜑 → [ ( ℩ 𝑥 𝜑 ) / 𝑥 ] 𝜑 )
2		df-sbc	⊢ ( [ ( ℩ 𝑥 𝜑 ) / 𝑥 ] 𝜑 ↔ ( ℩ 𝑥 𝜑 ) ∈ { 𝑥 ∣ 𝜑 } )
3	1 2	sylib	⊢ ( ∃! 𝑥 𝜑 → ( ℩ 𝑥 𝜑 ) ∈ { 𝑥 ∣ 𝜑 } )