Theorem an31 644

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 643	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ (𝜒 ∧ (𝜓 ∧ 𝜑)))
2		anass 468	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒)))
3		anass 468	. 2 ⊢ (((𝜒 ∧ 𝜓) ∧ 𝜑) ↔ (𝜒 ∧ (𝜓 ∧ 𝜑)))
4	1, 2, 3	3bitr4i 302	1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜒 ∧ 𝜓) ∧ 𝜑))

Syntax hints:   ↔ wb 205   ∧ wa 395

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

This theorem is referenced by:  euind  3654  reuind  3683  dchrelbas3  26291  lhpexle3  37953  4an31  42007  abciffcbatnabciffncba  44311