Theorem an13 840

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

Assertion

Ref	Expression
an13	⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ (𝜒 ∧ (𝜓 ∧ 𝜑)))

Proof of Theorem an13

Step	Hyp	Ref	Expression
1		an12 838	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))
2		anass 681	. 2 ⊢ (((𝜓 ∧ 𝜑) ∧ 𝜒) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))
3		ancom 466	. 2 ⊢ (((𝜓 ∧ 𝜑) ∧ 𝜒) ↔ (𝜒 ∧ (𝜓 ∧ 𝜑)))
4	1, 2, 3	3bitr2i 288	1 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ (𝜒 ∧ (𝜓 ∧ 𝜑)))

Syntax hints:   ↔ wb 196   ∧ wa 384

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 197  df-an 386

This theorem is referenced by:  an31  841  elxp2OLD  5133  elsnxp  5677  elsnxpOLD  5678  dchrelbas3  24963  dfiota3  32030  bj-dfmpt2a  33071  islpln5  34821  islvol5  34865  dibelval3  36436  opeliun2xp  42111