Theorem 3anan12 990

Description: Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
3anan12	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))

Proof of Theorem 3anan12

Step	Hyp	Ref	Expression
1		3ancoma 985	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜑 ∧ 𝜒))
2		3anass 982	. 2 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))
3	1, 2	bitri 184	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∧ w3a 978

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 980

This theorem is referenced by:  fncnv  5284  dff1o2  5468