Theorem 3anan12 1101

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 1103 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 1100	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒)))
2		an12 651	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))
3	1, 2	bitri 276	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))

Syntax hints:   ↔ wb 207   ∧ wa 396   ∧ w3a 1092

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 208  df-an 397  df-3an 1094

This theorem is referenced by:  3ancoma  1103  an33rean  1491  2reu5lem3  3698  snopeqop  5447  dff1o2  6772  ixxun  13305  elfz1b  13538  mreexexlem4d  17604  unocv  21655  iunocv  21656  iscvsp  25113  mbfmax  25634  ulm2  26368  iswwlks  29922  wwlksnfi  29992  eclclwwlkn1  30163  clwwlknon2x  30191  bnj548  35079  pridlnr  38403  brres2  38640  xrninxp  38782  sineq0ALT  45380  elbigo  49042