Theorem bigolden 972

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

Assertion

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

Proof of Theorem bigolden

Step	Hyp	Ref	Expression
1		pm4.71 660	. 2 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (𝜑 ∧ 𝜓)))
2		pm4.72 916	. 2 ⊢ ((𝜑 → 𝜓) ↔ (𝜓 ↔ (𝜑 ∨ 𝜓)))
3		bicom 211	. 2 ⊢ ((𝜑 ↔ (𝜑 ∧ 𝜓)) ↔ ((𝜑 ∧ 𝜓) ↔ 𝜑))
4	1, 2, 3	3bitr3ri 290	1 ⊢ (((𝜑 ∧ 𝜓) ↔ 𝜑) ↔ (𝜓 ↔ (𝜑 ∨ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385

This theorem is referenced by: (None)