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Theorem 3anan32 1097
Description: Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
Assertion
Ref Expression
3anan32 ((𝜑𝜓𝜒) ↔ ((𝜑𝜒) ∧ 𝜓))

Proof of Theorem 3anan32
StepHypRef Expression
1 df-3an 1089 . 2 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
2 an32 644 . 2 (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∧ 𝜓))
31, 2bitri 274 1 ((𝜑𝜓𝜒) ↔ ((𝜑𝜒) ∧ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  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 206  df-an 397  df-3an 1089
This theorem is referenced by:  3ancomb  1099  anandi3r  1103  rabssrabd  4040  dff1o3  6788  bropfvvvvlem  8020  tz7.49c  8389  ispos2  18201  lbsacsbs  20613  obslbs  21132  islbs4  21234  leordtvallem1  22557  trfbas2  23190  isclmp  24456  lssbn  24712  sineq0  25876  dchrelbas3  26582  elno3  26999  nb3grpr2  28229  uspgr2wlkeq  28492  2spthd  28784  clwwlknonwwlknonb  28948  frgr2wwlkeu  29169  elicoelioo  31576  cndprobprob  32929  bnj543  33396  cusgr3cyclex  33621  ellimits  34484  refsymrel2  37018  refsymrel3  37019  dfeqvrel2  37041  dfeqvrel3  37042  i0oii  46922
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