Theorem iota4

Description: Theorem *14.22 in WhiteheadRussell p. 190. (Contributed by Andrew Salmon, 12-Jul-2011)

	Ref	Expression
Assertion	iota4	$$⊢\exists !x\phi \to \underset{\dot{}}{[}\left(\iota x|\phi \right)/x\underset{\dot{}}{]}\phi$$

Proof

Step	Hyp	Ref	Expression
1		eu6	$$⊢\exists !x\phi \leftrightarrow \exists z\forall x\left(\phi \leftrightarrow x=z\right)$$
2		biimpr	$$⊢\left(\phi \leftrightarrow x=z\right)\to \left(x=z\to \phi \right)$$
3	2	alimi	$$⊢\forall x\left(\phi \leftrightarrow x=z\right)\to \forall x\left(x=z\to \phi \right)$$
4		sb6	$$⊢\left[z/x\right]\phi \leftrightarrow \forall x\left(x=z\to \phi \right)$$
5	3 4	sylibr	$$⊢\forall x\left(\phi \leftrightarrow x=z\right)\to \left[z/x\right]\phi$$
6		iotaval	$$⊢\forall x\left(\phi \leftrightarrow x=z\right)\to \left(\iota x|\phi \right)=z$$
7	6	eqcomd	$$⊢\forall x\left(\phi \leftrightarrow x=z\right)\to z=\left(\iota x|\phi \right)$$
8		dfsbcq2	$$⊢z=\left(\iota x|\phi \right)\to \left(\left[z/x\right]\phi \leftrightarrow \underset{\dot{}}{[}\left(\iota x|\phi \right)/x\underset{\dot{}}{]}\phi \right)$$
9	7 8	syl	$$⊢\forall x\left(\phi \leftrightarrow x=z\right)\to \left(\left[z/x\right]\phi \leftrightarrow \underset{\dot{}}{[}\left(\iota x|\phi \right)/x\underset{\dot{}}{]}\phi \right)$$
10	5 9	mpbid	$$⊢\forall x\left(\phi \leftrightarrow x=z\right)\to \underset{\dot{}}{[}\left(\iota x|\phi \right)/x\underset{\dot{}}{]}\phi$$
11	10	exlimiv	$$⊢\exists z\forall x\left(\phi \leftrightarrow x=z\right)\to \underset{\dot{}}{[}\left(\iota x|\phi \right)/x\underset{\dot{}}{]}\phi$$
12	1 11	sylbi	$$⊢\exists !x\phi \to \underset{\dot{}}{[}\left(\iota x|\phi \right)/x\underset{\dot{}}{]}\phi$$