Theorem 3an6 1262

Description: Analog of an4 797 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
3an6	⊢ (((φ ∧ ψ) ∧ (χ ∧ θ) ∧ (τ ∧ η)) ↔ ((φ ∧ χ ∧ τ) ∧ (ψ ∧ θ ∧ η)))

Proof of Theorem 3an6

Step	Hyp	Ref	Expression
1		an6 1261	. 2 ⊢ (((φ ∧ χ ∧ τ) ∧ (ψ ∧ θ ∧ η)) ↔ ((φ ∧ ψ) ∧ (χ ∧ θ) ∧ (τ ∧ η)))
2	1	bicomi 193	1 ⊢ (((φ ∧ ψ) ∧ (χ ∧ θ) ∧ (τ ∧ η)) ↔ ((φ ∧ χ ∧ τ) ∧ (ψ ∧ θ ∧ η)))

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:  sfin112  4530  sfinltfin  4536