Theorem bigolden

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

	Ref	Expression
Assertion	bigolden	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ↔ 𝜑 ) ↔ ( 𝜓 ↔ ( 𝜑 ∨ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pm4.71	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 ↔ ( 𝜑 ∧ 𝜓 ) ) )
2		pm4.72	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜓 ↔ ( 𝜑 ∨ 𝜓 ) ) )
3		bicom	⊢ ( ( 𝜑 ↔ ( 𝜑 ∧ 𝜓 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ↔ 𝜑 ) )
4	1 2 3	3bitr3ri	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ↔ 𝜑 ) ↔ ( 𝜓 ↔ ( 𝜑 ∨ 𝜓 ) ) )