Theorem pm10.542

Description: Theorem *10.542 in WhiteheadRussell p. 156. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	pm10.542	$$⊢\forall x\left(\phi \to \left(\chi \to \psi \right)\right)\leftrightarrow \left(\chi \to \forall x\left(\phi \to \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		bi2.04	$$⊢\left(\phi \to \left(\chi \to \psi \right)\right)\leftrightarrow \left(\chi \to \left(\phi \to \psi \right)\right)$$
2	1	albii	$$⊢\forall x\left(\phi \to \left(\chi \to \psi \right)\right)\leftrightarrow \forall x\left(\chi \to \left(\phi \to \psi \right)\right)$$
3		19.21v	$$⊢\forall x\left(\chi \to \left(\phi \to \psi \right)\right)\leftrightarrow \left(\chi \to \forall x\left(\phi \to \psi \right)\right)$$
4	2 3	bitri	$$⊢\forall x\left(\phi \to \left(\chi \to \psi \right)\right)\leftrightarrow \left(\chi \to \forall x\left(\phi \to \psi \right)\right)$$