Theorem ifor

Description: Rewrite a disjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014)

	Ref	Expression
Assertion	ifor	$$⊢if\left(\left(\phi \vee \psi \right),A,B\right)=if\left(\phi ,A,if\left(\psi ,A,B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		iftrue	$$⊢\left(\phi \vee \psi \right)\to if\left(\left(\phi \vee \psi \right),A,B\right)=A$$
2	1	orcs	$$⊢\phi \to if\left(\left(\phi \vee \psi \right),A,B\right)=A$$
3		iftrue	$$⊢\phi \to if\left(\phi ,A,if\left(\psi ,A,B\right)\right)=A$$
4	2 3	eqtr4d	$$⊢\phi \to if\left(\left(\phi \vee \psi \right),A,B\right)=if\left(\phi ,A,if\left(\psi ,A,B\right)\right)$$
5		iffalse	$$⊢\neg \phi \to if\left(\phi ,A,if\left(\psi ,A,B\right)\right)=if\left(\psi ,A,B\right)$$
6		biorf	$$⊢\neg \phi \to \left(\psi \leftrightarrow \left(\phi \vee \psi \right)\right)$$
7	6	ifbid	$$⊢\neg \phi \to if\left(\psi ,A,B\right)=if\left(\left(\phi \vee \psi \right),A,B\right)$$
8	5 7	eqtr2d	$$⊢\neg \phi \to if\left(\left(\phi \vee \psi \right),A,B\right)=if\left(\phi ,A,if\left(\psi ,A,B\right)\right)$$
9	4 8	pm2.61i	$$⊢if\left(\left(\phi \vee \psi \right),A,B\right)=if\left(\phi ,A,if\left(\psi ,A,B\right)\right)$$