Theorem 4an21 44649

Description: Rearrangement of 4 conjuncts with a triple conjunction. (Contributed by AV, 4-Mar-2022.)

Assertion

Ref	Expression
4an21	⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒 ∧ 𝜃)))

Proof of Theorem 4an21

Step	Hyp	Ref	Expression
1		3anass 1093	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
2		ancom 460	. . . 4 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
3	2	anbi1i 623	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜓 ∧ 𝜑) ∧ (𝜒 ∧ 𝜃)))
4		anass 468	. . . 4 ⊢ (((𝜓 ∧ 𝜑) ∧ (𝜒 ∧ 𝜃)) ↔ (𝜓 ∧ (𝜑 ∧ (𝜒 ∧ 𝜃))))
5		3anass 1093	. . . . . 6 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ (𝜒 ∧ 𝜃)))
6	5	bicomi 223	. . . . 5 ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜃)) ↔ (𝜑 ∧ 𝜒 ∧ 𝜃))
7	6	anbi2i 622	. . . 4 ⊢ ((𝜓 ∧ (𝜑 ∧ (𝜒 ∧ 𝜃))) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒 ∧ 𝜃)))
8	4, 7	bitri 274	. . 3 ⊢ (((𝜓 ∧ 𝜑) ∧ (𝜒 ∧ 𝜃)) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒 ∧ 𝜃)))
9	3, 8	bitri 274	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒 ∧ 𝜃)))
10	1, 9	bitri 274	1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒 ∧ 𝜃)))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1085

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 396  df-3an 1087

This theorem is referenced by: (None)