Theorem rb-ax1

Description: The first of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	rb-ax1	$$⊢\left(\neg \left(\neg \psi \vee \chi \right)\vee \left(\neg \left(\phi \vee \psi \right)\vee \left(\phi \vee \chi \right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		orim2	$$⊢\left(\psi \to \chi \right)\to \left(\left(\phi \vee \psi \right)\to \left(\phi \vee \chi \right)\right)$$
2		imor	$$⊢\left(\psi \to \chi \right)\leftrightarrow \left(\neg \psi \vee \chi \right)$$
3		imor	$$⊢\left(\left(\phi \vee \psi \right)\to \left(\phi \vee \chi \right)\right)\leftrightarrow \left(\neg \left(\phi \vee \psi \right)\vee \left(\phi \vee \chi \right)\right)$$
4	1 2 3	3imtr3i	$$⊢\left(\neg \psi \vee \chi \right)\to \left(\neg \left(\phi \vee \psi \right)\vee \left(\phi \vee \chi \right)\right)$$
5	4	imori	$$⊢\left(\neg \left(\neg \psi \vee \chi \right)\vee \left(\neg \left(\phi \vee \psi \right)\vee \left(\phi \vee \chi \right)\right)\right)$$