Theorem an6 1261

Description: Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.)

Assertion

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

Proof of Theorem an6

Step	Hyp	Ref	Expression
1		an4 797	. . 3 ⊢ ((((φ ∧ ψ) ∧ χ) ∧ ((θ ∧ τ) ∧ η)) ↔ (((φ ∧ ψ) ∧ (θ ∧ τ)) ∧ (χ ∧ η)))
2		an4 797	. . . 4 ⊢ (((φ ∧ ψ) ∧ (θ ∧ τ)) ↔ ((φ ∧ θ) ∧ (ψ ∧ τ)))
3	2	anbi1i 676	. . 3 ⊢ ((((φ ∧ ψ) ∧ (θ ∧ τ)) ∧ (χ ∧ η)) ↔ (((φ ∧ θ) ∧ (ψ ∧ τ)) ∧ (χ ∧ η)))
4	1, 3	bitri 240	. 2 ⊢ ((((φ ∧ ψ) ∧ χ) ∧ ((θ ∧ τ) ∧ η)) ↔ (((φ ∧ θ) ∧ (ψ ∧ τ)) ∧ (χ ∧ η)))
5		df-3an 936	. . 3 ⊢ ((φ ∧ ψ ∧ χ) ↔ ((φ ∧ ψ) ∧ χ))
6		df-3an 936	. . 3 ⊢ ((θ ∧ τ ∧ η) ↔ ((θ ∧ τ) ∧ η))
7	5, 6	anbi12i 678	. 2 ⊢ (((φ ∧ ψ ∧ χ) ∧ (θ ∧ τ ∧ η)) ↔ (((φ ∧ ψ) ∧ χ) ∧ ((θ ∧ τ) ∧ η)))
8		df-3an 936	. 2 ⊢ (((φ ∧ θ) ∧ (ψ ∧ τ) ∧ (χ ∧ η)) ↔ (((φ ∧ θ) ∧ (ψ ∧ τ)) ∧ (χ ∧ η)))
9	4, 7, 8	3bitr4i 268	1 ⊢ (((φ ∧ ψ ∧ χ) ∧ (θ ∧ τ ∧ η)) ↔ ((φ ∧ θ) ∧ (ψ ∧ τ) ∧ (χ ∧ 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:  3an6  1262  fntxp  5805