Theorem pm14.12

Description: Theorem *14.12 in WhiteheadRussell p. 184. (Contributed by Andrew Salmon, 11-Jul-2011)

	Ref	Expression
Assertion	pm14.12	$$⊢\exists !x\phi \to \forall x\forall y\left(\left(\phi \wedge \underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \right)\to x=y\right)$$

Proof

Step	Hyp	Ref	Expression
1		eumo	$$⊢\exists !x\phi \to {\exists }^{*}x\phi$$
2		nfv	$$⊢Ⅎy\phi$$
3	2	mo3	$$⊢{\exists }^{*}x\phi \leftrightarrow \forall x\forall y\left(\left(\phi \wedge \left[y/x\right]\phi \right)\to x=y\right)$$
4		sbsbc	$$⊢\left[y/x\right]\phi \leftrightarrow \underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi$$
5	4	anbi2i	$$⊢\left(\phi \wedge \left[y/x\right]\phi \right)\leftrightarrow \left(\phi \wedge \underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \right)$$
6	5	imbi1i	$$⊢\left(\left(\phi \wedge \left[y/x\right]\phi \right)\to x=y\right)\leftrightarrow \left(\left(\phi \wedge \underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \right)\to x=y\right)$$
7	6	2albii	$$⊢\forall x\forall y\left(\left(\phi \wedge \left[y/x\right]\phi \right)\to x=y\right)\leftrightarrow \forall x\forall y\left(\left(\phi \wedge \underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \right)\to x=y\right)$$
8	3 7	bitri	$$⊢{\exists }^{*}x\phi \leftrightarrow \forall x\forall y\left(\left(\phi \wedge \underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \right)\to x=y\right)$$
9	1 8	sylib	$$⊢\exists !x\phi \to \forall x\forall y\left(\left(\phi \wedge \underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \right)\to x=y\right)$$