Theorem bigolden 862

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		pm4.71 369	. 2 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (𝜑 ∧ 𝜓)))
2		pm4.72 736	. 2 ⊢ ((𝜑 → 𝜓) ↔ (𝜓 ↔ (𝜑 ∨ 𝜓)))
3		bicom 128	. 2 ⊢ ((𝜑 ↔ (𝜑 ∧ 𝜓)) ↔ ((𝜑 ∧ 𝜓) ↔ 𝜑))
4	1, 2, 3	3bitr3ri 200	1 ⊢ (((𝜑 ∧ 𝜓) ↔ 𝜑) ↔ (𝜓 ↔ (𝜑 ∨ 𝜓)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)