Theorem eupickbi

Description: Theorem *14.26 in WhiteheadRussell p. 192. (Contributed by Andrew Salmon, 11-Jul-2011) (Proof shortened by Wolf Lammen, 27-Dec-2018)

	Ref	Expression
Assertion	eupickbi	$$⊢\exists !x\phi \to \left(\exists x\left(\phi \wedge \psi \right)\leftrightarrow \forall x\left(\phi \to \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		eupicka	$$⊢\left(\exists !x\phi \wedge \exists x\left(\phi \wedge \psi \right)\right)\to \forall x\left(\phi \to \psi \right)$$
2	1	ex	$$⊢\exists !x\phi \to \left(\exists x\left(\phi \wedge \psi \right)\to \forall x\left(\phi \to \psi \right)\right)$$
3		euex	$$⊢\exists !x\phi \to \exists x\phi$$
4		exintr	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\exists x\phi \to \exists x\left(\phi \wedge \psi \right)\right)$$
5	3 4	syl5com	$$⊢\exists !x\phi \to \left(\forall x\left(\phi \to \psi \right)\to \exists x\left(\phi \wedge \psi \right)\right)$$
6	2 5	impbid	$$⊢\exists !x\phi \to \left(\exists x\left(\phi \wedge \psi \right)\leftrightarrow \forall x\left(\phi \to \psi \right)\right)$$