Theorem elimif

Description: Elimination of a conditional operator contained in a wff ps . (Contributed by NM, 15-Feb-2005) (Proof shortened by NM, 25-Apr-2019)

	Ref	Expression
Hypotheses	elimif.1	$$⊢if\left(\phi ,A,B\right)=A\to \left(\psi \leftrightarrow \chi \right)$$
	elimif.2	$$⊢if\left(\phi ,A,B\right)=B\to \left(\psi \leftrightarrow \theta \right)$$
Assertion	elimif	$$⊢\psi \leftrightarrow \left(\left(\phi \wedge \chi \right)\vee \left(\neg \phi \wedge \theta \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		elimif.1	$$⊢if\left(\phi ,A,B\right)=A\to \left(\psi \leftrightarrow \chi \right)$$
2		elimif.2	$$⊢if\left(\phi ,A,B\right)=B\to \left(\psi \leftrightarrow \theta \right)$$
3		iftrue	$$⊢\phi \to if\left(\phi ,A,B\right)=A$$
4	3 1	syl	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
5		iffalse	$$⊢\neg \phi \to if\left(\phi ,A,B\right)=B$$
6	5 2	syl	$$⊢\neg \phi \to \left(\psi \leftrightarrow \theta \right)$$
7	4 6	cases	$$⊢\psi \leftrightarrow \left(\left(\phi \wedge \chi \right)\vee \left(\neg \phi \wedge \theta \right)\right)$$