Theorem bigolden 513

Description: Dijkstra-Scholten's Golden Rule for calculational proofs.

Assertion

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

Proof of Theorem bigolden

Step	Hyp	Ref	Expression
1		pm4.71 481	. 2 ⊢ ((φ → ψ) ↔ (φ ↔ (φ ∧ ψ)))
2		pm4.72 485	. 2 ⊢ ((φ → ψ) ↔ (ψ ↔ (φ ∨ ψ)))
3		bicom 398	. 2 ⊢ ((φ ↔ (φ ∧ ψ)) ↔ ((φ ∧ ψ) ↔ φ))
4	1, 2, 3	3bitr3r 157	1 ⊢ (((φ ∧ ψ) ↔ φ) ↔ (ψ ↔ (φ ∨ ψ)))

Syntax hints:   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

metamath.org