Theorem bigolden

Description: Dijkstra-Scholten's Golden Rule for calculational proofs. (Contributed by NM, 10-Jan-2005)

	Ref	Expression
Assertion	bigolden	$$⊢\left(\left(\phi \wedge \psi \right)\leftrightarrow \phi \right)\leftrightarrow \left(\psi \leftrightarrow \left(\phi \vee \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		pm4.71	$$⊢\left(\phi \to \psi \right)\leftrightarrow \left(\phi \leftrightarrow \left(\phi \wedge \psi \right)\right)$$
2		pm4.72	$$⊢\left(\phi \to \psi \right)\leftrightarrow \left(\psi \leftrightarrow \left(\phi \vee \psi \right)\right)$$
3		bicom	$$⊢\left(\phi \leftrightarrow \left(\phi \wedge \psi \right)\right)\leftrightarrow \left(\left(\phi \wedge \psi \right)\leftrightarrow \phi \right)$$
4	1 2 3	3bitr3ri	$$⊢\left(\left(\phi \wedge \psi \right)\leftrightarrow \phi \right)\leftrightarrow \left(\psi \leftrightarrow \left(\phi \vee \psi \right)\right)$$