Theorem 3anan12 1096

Description: Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised to shorten 3ancoma 1098 by Wolf Lammen, 5-Jun-2022.)

Assertion

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

Proof of Theorem 3anan12

Step	Hyp	Ref	Expression
1		3anass 1095	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒)))
2		an12 646	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))
3	1, 2	bitri 275	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087

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 207  df-an 396  df-3an 1089

This theorem is referenced by:  3ancoma  1098  an33rean  1486  2reu5lem3  3704  snopeqop  5455  dff1o2  6780  ixxun  13308  elfz1b  13541  mreexexlem4d  17607  unocv  21673  iunocv  21674  iscvsp  25108  mbfmax  25629  ulm2  26366  iswwlks  29922  wwlksnfi  29992  eclclwwlkn1  30163  clwwlknon2x  30191  bnj548  35058  pridlnr  38374  brres2  38611  xrninxp  38753  sineq0ALT  45384  elbigo  49042