Theorem an31 553

Description: A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)

Assertion

Ref	Expression
an31	⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜒 ∧ 𝜓) ∧ 𝜑))

Proof of Theorem an31

Step	Hyp	Ref	Expression
1		an13 552	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ (𝜒 ∧ (𝜓 ∧ 𝜑)))
2		anass 398	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒)))
3		anass 398	. 2 ⊢ (((𝜒 ∧ 𝜓) ∧ 𝜑) ↔ (𝜒 ∧ (𝜓 ∧ 𝜑)))
4	1, 2, 3	3bitr4i 211	1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜒 ∧ 𝜓) ∧ 𝜑))

Syntax hints:   ∧ wa 103   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  euind  2871  reuind  2889