Theorem 3anan32 946

Description: Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Assertion

Ref	Expression
3anan32	⊢ ((φ ∧ ψ ∧ χ) ↔ ((φ ∧ χ) ∧ ψ))

Proof of Theorem 3anan32

Step	Hyp	Ref	Expression
1		df-3an 936	. 2 ⊢ ((φ ∧ ψ ∧ χ) ↔ ((φ ∧ ψ) ∧ χ))
2		an32 773	. 2 ⊢ (((φ ∧ ψ) ∧ χ) ↔ ((φ ∧ χ) ∧ ψ))
3	1, 2	bitri 240	1 ⊢ ((φ ∧ ψ ∧ χ) ↔ ((φ ∧ χ) ∧ ψ))

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358   ∧ w3a 934

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-an 360  df-3an 936

This theorem is used by: (None)