Theorem iotavalb

Description: Theorem *14.202 in WhiteheadRussell p. 189. A biconditional version of iotaval . (Contributed by Andrew Salmon, 11-Jul-2011)

	Ref	Expression
Assertion	iotavalb	$$⊢\exists !x\phi \to \left(\forall x\left(\phi \leftrightarrow x=y\right)\leftrightarrow \left(\iota x|\phi \right)=y\right)$$

Proof

Step	Hyp	Ref	Expression
1		iotaval	$$⊢\forall x\left(\phi \leftrightarrow x=y\right)\to \left(\iota x|\phi \right)=y$$
2		iotasbc	$$⊢\exists !x\phi \to \left(\underset{\dot{}}{[}\left(\iota x|\phi \right)/z\underset{\dot{}}{]}z=y\leftrightarrow \exists z\left(\forall x\left(\phi \leftrightarrow x=z\right)\wedge z=y\right)\right)$$
3		iotaexeu	$$⊢\exists !x\phi \to \left(\iota x|\phi \right)\in \mathrm{V}$$
4		eqsbc1	$$⊢\left(\iota x|\phi \right)\in \mathrm{V}\to \left(\underset{\dot{}}{[}\left(\iota x|\phi \right)/z\underset{\dot{}}{]}z=y\leftrightarrow \left(\iota x|\phi \right)=y\right)$$
5	3 4	syl	$$⊢\exists !x\phi \to \left(\underset{\dot{}}{[}\left(\iota x|\phi \right)/z\underset{\dot{}}{]}z=y\leftrightarrow \left(\iota x|\phi \right)=y\right)$$
6	2 5	bitr3d	$$⊢\exists !x\phi \to \left(\exists z\left(\forall x\left(\phi \leftrightarrow x=z\right)\wedge z=y\right)\leftrightarrow \left(\iota x|\phi \right)=y\right)$$
7		equequ2	$$⊢z=y\to \left(x=z\leftrightarrow x=y\right)$$
8	7	bibi2d	$$⊢z=y\to \left(\left(\phi \leftrightarrow x=z\right)\leftrightarrow \left(\phi \leftrightarrow x=y\right)\right)$$
9	8	albidv	$$⊢z=y\to \left(\forall x\left(\phi \leftrightarrow x=z\right)\leftrightarrow \forall x\left(\phi \leftrightarrow x=y\right)\right)$$
10	9	biimpac	$$⊢\left(\forall x\left(\phi \leftrightarrow x=z\right)\wedge z=y\right)\to \forall x\left(\phi \leftrightarrow x=y\right)$$
11	10	exlimiv	$$⊢\exists z\left(\forall x\left(\phi \leftrightarrow x=z\right)\wedge z=y\right)\to \forall x\left(\phi \leftrightarrow x=y\right)$$
12	6 11	syl6bir	$$⊢\exists !x\phi \to \left(\left(\iota x|\phi \right)=y\to \forall x\left(\phi \leftrightarrow x=y\right)\right)$$
13	1 12	impbid2	$$⊢\exists !x\phi \to \left(\forall x\left(\phi \leftrightarrow x=y\right)\leftrightarrow \left(\iota x|\phi \right)=y\right)$$