Theorem an6 1444

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 653	. . 3 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ ((𝜃 ∧ 𝜏) ∧ 𝜂)) ↔ (((𝜑 ∧ 𝜓) ∧ (𝜃 ∧ 𝜏)) ∧ (𝜒 ∧ 𝜂)))
2		an4 653	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜃 ∧ 𝜏)) ↔ ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜏)))
3	2	anbi1i 624	. . 3 ⊢ ((((𝜑 ∧ 𝜓) ∧ (𝜃 ∧ 𝜏)) ∧ (𝜒 ∧ 𝜂)) ↔ (((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜏)) ∧ (𝜒 ∧ 𝜂)))
4	1, 3	bitri 274	. 2 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ ((𝜃 ∧ 𝜏) ∧ 𝜂)) ↔ (((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜏)) ∧ (𝜒 ∧ 𝜂)))
5		df-3an 1088	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
6		df-3an 1088	. . 3 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜂) ↔ ((𝜃 ∧ 𝜏) ∧ 𝜂))
7	5, 6	anbi12i 627	. 2 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏 ∧ 𝜂)) ↔ (((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ ((𝜃 ∧ 𝜏) ∧ 𝜂)))
8		df-3an 1088	. 2 ⊢ (((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜏) ∧ (𝜒 ∧ 𝜂)) ↔ (((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜏)) ∧ (𝜒 ∧ 𝜂)))
9	4, 7, 8	3bitr4i 303	1 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏 ∧ 𝜂)) ↔ ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜏) ∧ (𝜒 ∧ 𝜂)))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088

This theorem is referenced by:  3an6  1445  elfzuzb  13250  fzadd2  13291  ptbasin  22728  iimulcl  24100  nb3grpr  27749  nb3grpr2  27750  txpconn  33194  paddasslem9  37842  paddasslem10  37843  gboge9  45216