Theorem 3anan12 1112

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 1115 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 1111	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒)))
2		an12 658	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))
3	1, 2	bitri 278	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105

This theorem is used by:  3anan32  1113  3ancoma  1115  an33rean  1514  2reu5lem3  3722  snopeqop  5491  dff1o2  6830  ixxun  13406  elfz1b  13640  mreexexlem4d  17727  unocv  21882  iunocv  21883  iscvsp  25340  mbfmax  25861  ulm2  26601  iswwlks  30254  wwlksnfi  30324  eclclwwlkn1  30495  clwwlknon2x  30523  bnj548  35352  pridlnr  38747  brres2  38982  xrninxp  39124  sineq0ALT  45705  elbigo  49390