Theorem pm10.541

Description: Theorem *10.541 in WhiteheadRussell p. 155. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	pm10.541	$$⊢\forall x\left(\phi \to \left(\chi \vee \psi \right)\right)\leftrightarrow \left(\chi \vee \forall x\left(\phi \to \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		bi2.04	$$⊢\left(\phi \to \left(\neg \chi \to \psi \right)\right)\leftrightarrow \left(\neg \chi \to \left(\phi \to \psi \right)\right)$$
2	1	albii	$$⊢\forall x\left(\phi \to \left(\neg \chi \to \psi \right)\right)\leftrightarrow \forall x\left(\neg \chi \to \left(\phi \to \psi \right)\right)$$
3		19.21v	$$⊢\forall x\left(\neg \chi \to \left(\phi \to \psi \right)\right)\leftrightarrow \left(\neg \chi \to \forall x\left(\phi \to \psi \right)\right)$$
4	2 3	bitri	$$⊢\forall x\left(\phi \to \left(\neg \chi \to \psi \right)\right)\leftrightarrow \left(\neg \chi \to \forall x\left(\phi \to \psi \right)\right)$$
5		df-or	$$⊢\left(\chi \vee \psi \right)\leftrightarrow \left(\neg \chi \to \psi \right)$$
6	5	imbi2i	$$⊢\left(\phi \to \left(\chi \vee \psi \right)\right)\leftrightarrow \left(\phi \to \left(\neg \chi \to \psi \right)\right)$$
7	6	albii	$$⊢\forall x\left(\phi \to \left(\chi \vee \psi \right)\right)\leftrightarrow \forall x\left(\phi \to \left(\neg \chi \to \psi \right)\right)$$
8		df-or	$$⊢\left(\chi \vee \forall x\left(\phi \to \psi \right)\right)\leftrightarrow \left(\neg \chi \to \forall x\left(\phi \to \psi \right)\right)$$
9	4 7 8	3bitr4i	$$⊢\forall x\left(\phi \to \left(\chi \vee \psi \right)\right)\leftrightarrow \left(\chi \vee \forall x\left(\phi \to \psi \right)\right)$$