Theorem an4com24 43824

Description: Rearrangement of 4 conjuncts: second and forth positions interchanged. (Contributed by AV, 18-Feb-2022.)

Assertion

Ref	Expression
an4com24	⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜃) ∧ (𝜒 ∧ 𝜓)))

Proof of Theorem an4com24

Step	Hyp	Ref	Expression
1		an43 657	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒)))
2		ancom 464	. . 3 ⊢ ((𝜓 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓))
3	2	anbi2i 625	. 2 ⊢ (((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜃) ∧ (𝜒 ∧ 𝜓)))
4	1, 3	bitri 278	1 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜃) ∧ (𝜒 ∧ 𝜓)))

Syntax hints:   ↔ wb 209   ∧ wa 399

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 210  df-an 400

This theorem is referenced by:  3an4ancom24  43825