Theorem bigolden 901

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		pm4.71 611	. 2 ⊢ ((φ → ψ) ↔ (φ ↔ (φ ∧ ψ)))
2		pm4.72 846	. 2 ⊢ ((φ → ψ) ↔ (ψ ↔ (φ ∨ ψ)))
3		bicom 191	. 2 ⊢ ((φ ↔ (φ ∧ ψ)) ↔ ((φ ∧ ψ) ↔ φ))
4	1, 2, 3	3bitr3ri 267	1 ⊢ (((φ ∧ ψ) ↔ φ) ↔ (ψ ↔ (φ ∨ ψ)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360

This theorem is used by: (None)