Theorem riotasbc

Description: Substitution law for descriptions. Compare iotasbc . (Contributed by NM, 23-Aug-2011) (Proof shortened by Mario Carneiro, 24-Dec-2016)

	Ref	Expression
Assertion	riotasbc	$$⊢\exists !x\in A\phi \to \underset{\dot{}}{[}\left(\iota x\in A|\phi \right)/x\underset{\dot{}}{]}\phi$$

Proof

Step	Hyp	Ref	Expression
1		rabssab	$$⊢\left\{x\in A|\phi \right\}\subseteq \left\{x|\phi \right\}$$
2		riotacl2	$$⊢\exists !x\in A\phi \to \left(\iota x\in A|\phi \right)\in \left\{x\in A|\phi \right\}$$
3	1 2	sselid	$$⊢\exists !x\in A\phi \to \left(\iota x\in A|\phi \right)\in \left\{x|\phi \right\}$$
4		df-sbc	$$⊢\underset{\dot{}}{[}\left(\iota x\in A|\phi \right)/x\underset{\dot{}}{]}\phi \leftrightarrow \left(\iota x\in A|\phi \right)\in \left\{x|\phi \right\}$$
5	3 4	sylibr	$$⊢\exists !x\in A\phi \to \underset{\dot{}}{[}\left(\iota x\in A|\phi \right)/x\underset{\dot{}}{]}\phi$$