Theorem an6 638

Description: Rearrangement of 6 conjuncts.

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		df-3an 583	. . . 4 ⊢ ((φ ∧ ψ ∧ χ) ↔ ((φ ∧ ψ) ∧ χ))
2		df-3an 583	. . . 4 ⊢ ((θ ∧ τ ∧ η) ↔ ((θ ∧ τ) ∧ η))
3	1, 2	anbi12i 369	. . 3 ⊢ (((φ ∧ ψ ∧ χ) ∧ (θ ∧ τ ∧ η)) ↔ (((φ ∧ ψ) ∧ χ) ∧ ((θ ∧ τ) ∧ η)))
4		an4 388	. . 3 ⊢ ((((φ ∧ ψ) ∧ χ) ∧ ((θ ∧ τ) ∧ η)) ↔ (((φ ∧ ψ) ∧ (θ ∧ τ)) ∧ (χ ∧ η)))
5		an4 388	. . . 4 ⊢ (((φ ∧ ψ) ∧ (θ ∧ τ)) ↔ ((φ ∧ θ) ∧ (ψ ∧ τ)))
6	5	anbi1i 368	. . 3 ⊢ ((((φ ∧ ψ) ∧ (θ ∧ τ)) ∧ (χ ∧ η)) ↔ (((φ ∧ θ) ∧ (ψ ∧ τ)) ∧ (χ ∧ η)))
7	3, 4, 6	3bitr 155	. 2 ⊢ (((φ ∧ ψ ∧ χ) ∧ (θ ∧ τ ∧ η)) ↔ (((φ ∧ θ) ∧ (ψ ∧ τ)) ∧ (χ ∧ η)))
8		df-3an 583	. 2 ⊢ (((φ ∧ θ) ∧ (ψ ∧ τ) ∧ (χ ∧ η)) ↔ (((φ ∧ θ) ∧ (ψ ∧ τ)) ∧ (χ ∧ η)))
9	7, 8	bitr4 154	1 ⊢ (((φ ∧ ψ ∧ χ) ∧ (θ ∧ τ ∧ η)) ↔ ((φ ∧ θ) ∧ (ψ ∧ τ) ∧ (χ ∧ η)))

Syntax hints:   ↔ wb 127   ∧ wa 196   ∧ w3a 581

This theorem is referenced by:  f1oco 2816  distrlem3pr 3923

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-an 198  df-3an 583

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