Theorem pm11.62

Description: Theorem *11.62 in WhiteheadRussell p. 166. Importation combined with the rearrangement with quantifiers. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	pm11.62	$$⊢\forall x\forall y\left(\left(\phi \wedge \psi \right)\to \chi \right)\leftrightarrow \forall x\left(\phi \to \forall y\left(\psi \to \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		impexp	$$⊢\left(\left(\phi \wedge \psi \right)\to \chi \right)\leftrightarrow \left(\phi \to \left(\psi \to \chi \right)\right)$$
2	1	albii	$$⊢\forall y\left(\left(\phi \wedge \psi \right)\to \chi \right)\leftrightarrow \forall y\left(\phi \to \left(\psi \to \chi \right)\right)$$
3		19.21v	$$⊢\forall y\left(\phi \to \left(\psi \to \chi \right)\right)\leftrightarrow \left(\phi \to \forall y\left(\psi \to \chi \right)\right)$$
4	2 3	bitri	$$⊢\forall y\left(\left(\phi \wedge \psi \right)\to \chi \right)\leftrightarrow \left(\phi \to \forall y\left(\psi \to \chi \right)\right)$$
5	4	albii	$$⊢\forall x\forall y\left(\left(\phi \wedge \psi \right)\to \chi \right)\leftrightarrow \forall x\left(\phi \to \forall y\left(\psi \to \chi \right)\right)$$