Theorem 4anpull2 1362

Description: An equivalence of two four-terms conjunctions with the terms regrouped (here, the second sub-conjunct of the first term is pulled separately). (Contributed by Zhi Wang, 4-Sep-2024.)

Assertion

Ref	Expression
4anpull2	⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜓))

Proof of Theorem 4anpull2

Step	Hyp	Ref	Expression
1		anass 468	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ (𝜑 ∧ (𝜓 ∧ (𝜒 ∧ 𝜃))))
2		anass 468	. . 3 ⊢ (((𝜑 ∧ (𝜒 ∧ 𝜃)) ∧ 𝜓) ↔ (𝜑 ∧ ((𝜒 ∧ 𝜃) ∧ 𝜓)))
3		3anass 1095	. . . 4 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ (𝜒 ∧ 𝜃)))
4	3	anbi1i 624	. . 3 ⊢ (((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜓) ↔ ((𝜑 ∧ (𝜒 ∧ 𝜃)) ∧ 𝜓))
5		ancom 460	. . . 4 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜒 ∧ 𝜃) ∧ 𝜓))
6	5	anbi2i 623	. . 3 ⊢ ((𝜑 ∧ (𝜓 ∧ (𝜒 ∧ 𝜃))) ↔ (𝜑 ∧ ((𝜒 ∧ 𝜃) ∧ 𝜓)))
7	2, 4, 6	3bitr4ri 304	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ (𝜒 ∧ 𝜃))) ↔ ((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜓))
8	1, 7	bitri 275	1 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜓))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087

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 207  df-an 396  df-3an 1089

This theorem is referenced by:  evenwodadd  46876  iscnrm3lem4  48806